Heuristic Considerations On Fast Axopodial

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(O) e -Ixll "0 ,
V(x)
(1)
(2)
where V = potential across the membrane (mV), x = position along the length of the cable (cm),
"0 =length
constant (cm), d
= cable
diameter (cm),
Rm
= membrane
resistance (0 cm 2), and
Ri = resistivity of cytoplasm (0 cm) (Nossal and Lecar, 1991, p. 249). Note that
over which the membrane potential falls to
i of its initial value.
That is, V("o)
"0 is the distance
= V~O)
. Shepherd
"0 vs. d for three values of ~~ appropriate for unmyelinated
In the sequel, I read "0 from the middle curve of Fig. 13.
(1974, Fig. 13, p. 64) gives a graph of
1
mammalian nerve processes.
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 cm 2 for Paramecium, and 73 kO cm 2 for Stentor
(summarized by Wood, 1982, p. 538). [By comparison, for smooth muscle, Rm
al., 1983, p. 23).] Typically, Ri = 100 0 cm
(Jack et
kO cm (Jack et al., 1983, p.23), and for axopodia
Substituting these values in equation (2), we find that for
am = 14.3 kO
"0 = 2 x 102-2 x 103 JJm, whereas for Rm =73 kO cm2, ~o = 4 x 102_4 x 103 JJm.
Observe that
d = 1 x 10-5_1 x 10-3 cm.
cm 2,
= 0.1
= 40 kO cm 2
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
~
I
are close to those for mammalian neural processes, then Ao appears to be
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
Note that if AO
For Ao
~
~
~
Lo or AO
~
Lo ' where Lo is the initial (uncontracted) length of the axopodium.
Lo ' regeneration would be necessary if an action potential were to serve as a trigger.
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
)2
2F2z2c •
A - ( £RT
D -
(1)
where AD is the Debye length (m), R is the gas constant (8.314 .J K-Imol- l ), T is the absolute
temperature (K), F is the Faraday constant (9.648 x 10 4 C mor l ), z is the valence of the ion in
question, c is its concentration (mol m- 3 ), and
£
is the dielectri<; constant (permittivity) of water. Note
10
that (=
(O(r'
where
(0
is the permittivity of a vacuum ( = 8.851
relative permittivity of water. At 25° C (
= 298 K), = 78.3.
(r
X
10- 12 C 2
J'r1 m 2),
and
(r
is the
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
(Probstein, 1989, p. 187, eqn. 6.4.6).
(2)
Probstein observes that "for a univalent electrolyte the Debye
length is thus about 1 nm for a concentration of 10 2 mol m- 3 and 10 nm for 1 mol m- 3 ."
Now I compute approximately t.he Debye length of a double layer due to Ca 2 + on the surface of
H. marina and C. marma.
mM
= 10
mol m- 3 .
Taking z
Davidson (1975) states that for the seawater used, c = [Ca 2 +J = 10
=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.
c
How does the Debye length calculation turn out for Na +? For Na +, appropriate values are
= 460 mM = 460 mol m- 3 and z = 1.
3.
ions?
Inserting these data in equation (2) affords '\0
= 0.45 nm.
How can one calculate the Debye length for a double layer involving a realistic mixture of
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), Ca 2 + (10 mM), and Mg 2 + (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
4.
lower bound.
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 voltagegated Ca 2+ 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
solutions.
what is meant by electrically neutral
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 = ¢J w 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-H iickel 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 Ca 2+ 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
3.
~~ = 0, where z is the axial coordinate.
Diffusion occurs in a two-dimensional annulus such that the inner aspect of the axopodial
membrane is the source of Ca 2+, 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----
4.
A
----I
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
T
where D is the diffusion coefficient
A2 (A)
a ,
(1)
= 8D In
For Ca2+,
(Hardt, 1981).
typical published values of D in
cytoplasm are 6 x 10-10 m 2 s-1 (Hodgkin and Keynes, 1957), 4 x 10- 10 m 2 s-1 (Kargacin and Fay, 1991),
and 3 x 10-10 m 2 s-1
DCa = 3.8 X 10- 11 m 2 s-1
(Roberts, 1993, p. 74).
However, a recent determination reports
(Albritton et al., 1992).
Arbitrarily, I shall take 5 x 10-10 m 2 s-1 as a
middling compromise "classical" value of DCa' and I shall call 3.8 x 10- 11 m 2 s-1 the "revised" value.
For IP 3, Albritton et al. (1992) found D1P3 = 2.83 X 10- 10 m 2 s-l. Measurements of a and A are seldom
reported in the literature.
Echinosphaerium nucleolilum
Hoffman et al. (1983, p. 375) found that isolated axonemes measured
which they maintain is consistent with electron micrographs by Tilney.
~
1.5 pm in diameter,
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
T
=10 pm in equation (1), we obtain
= 5 X 10 1 ms.
Similarly, for a = 1.5 pm and A = 5 pm,
T
Hence
= 7 ms.
15
7 ms ~
T
~ 5 X 10 1 ms (classical).
Repeating the foregoing calculations with DCa = 3.8 X 10- 11 m 2 s-l yields
9 X 101 ms ~
T
~ 6 x 10 2 ms (revised).
For IP3' equation (1) affords
Echinosphaerium aka mae
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.
magnification factor is x 33,000, 1 cm
<-+
Since the
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
(3.6 X 10<6 m)2 In (12)
- 8(5 x 10- 10 m2 s<l)
3
T _
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:<oneme image diameter is 6 em, the
axopodium image diameter is 14 cm, and the magnification is x 88,000.
a
= 0.7 pm, A = 1.6 pm, and
Hence 1 cm
<-+
O.lIpm,
16
T
= 0.54 ms.
It follows that
~ T ~
0.54 ms
4.5 ms (classical).
Similar calculations using the revised value of DCa yield the range of times
7 ms ~
T
~ 6 X 10 1 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
and A = 0.1 J.lm,
7
(1 X 10- m)2 In (100)
- 8(5 x 10- 10 m 2 s·l)
50
T _
T
= 9 J.lS (classical).
The same calculation with DCa = 3.8 X 10- 11 m 2 s-1 gives
T
= 1.2 x 102
J.ls (revised).
= 50 nm
17
For IP 3,
TIP
3
= 16 J.ls.
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 10 2 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
discrepancies are harder to reconcile than for Ca2+ because IP 3 derives from the hydrolysis of a plasma
membrane-bound precursor, phosphatidylinositol 4,5-bisphosphate (PIP 2), and is not released from
cytoplasmic stores.
In Echinosphaerium, 0.9 ms ~ TI?3 ~ 8 X 10 1 ms.
Observe that the even smaller
diffusion coefficient of ATP in cytoplasm (D AT ? = 1.50 x 10- 10 m 2 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 IP 3 , 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
dc
- 0, r -- a
dr c(O,t) <
00
'v't > o.
The solution is
where ai > 0, J 1(ai)
== 0, and J k( ) is the kth 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.
00
D; for values pertinent to diffusion in muscle.
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
by rCa 2+ 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 [Ca 2+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
found a statement of the threshold [Ca 2+l e .)
Axopodia shorten slowlv in the presence of the divalent cation ionophore A23187 (1 J1.g mr 1)
and Ca 2+. (Time to shorten to 49% of extended length = 10 min.) A much smaller extent of
shortening was seen at lower [Ca 2+l e.
Echinosphaerium eichhorni (Schliwa, 1976, 1977)
Slow contraction was observed for [Ca2 +l e as low as 10- 5 M in the presence of 0.5 J1.g mr 1
A23187. The speed of contraction increases with [Ca 2 +l e : half-maximal contraction was observed
after 35 min in 1 x 10- 4 M Ca 2 + and 12 min in 5 x 10- 4 1\1 Ca2 +. In tile most concentrated
solution tried ([Ca 2+]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 Ca 2+ 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 Ca 2+. The threshold concentration for
contraction is 5 x 10- 7 M. Stalks remain contrilctcd ill. [Ca 2+]e = 10- 6 I\L The external work
done in contraction in vivo can be accounted for hy a change of [Ca 2 +]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 [Ca 2 +]e = 10- 8 1\1, and contracted with stalk
flexure in 10- 5 M Ca 2+
"a,<;
in life but taking several seconds and without suhsequent relaxation"
(p. 303). The isolated spasmoneme shows arpropriate hehavior at the same values of [Ca 2+]e'
Ciliate Cell Body Contraction
B/epharisma japonicllm (Matsuoka et aI., 1991)
Results for Triton-extracted models:
The threshold concentration for model elongation is [Ca 2+]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 [Ca 2+]e = 10- 0 1\1 (without ATP).
Stentor (Huang and Mazia, 1975, pp. 393-391)
Results for EGT A-treated cells:
Cells remain extended in 10- 8 1\1 Ca 2 +.
Cells contract in 10- 6 M Ca 2+.
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 [Ca 2+]e
< 10- 6 ~'l. but swim backward or tumble for [Ca 2+]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 [Ca 2 +]e < 10- 6 ~I, backward swimming for
[Ca 2 +]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 [Ca 2 +]e:::; 10- 7 M, backward swimming for [Ca 2 +]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 Ca 2+.
Ceratium tripos (Maruyama, 1985; H6hfeld et. aI., 1988)
Results on reactivated longitudinal nagella:
Contraction requires [Ca 2+]e ~ ] x 10- 5 l\L
Oxyrrhis marina (Godart and Huitorel. ]902)
Results on longitudinal flagella still attached to permeabilized cells:
[Ca 2 +]e = 1 x 10- 7 M
50% contraction
[Ca 2+]e = 3 x 10- 7 M
50% autotomy
[9a2+]e = 9 x 10- 7 M
00% autot.omy
26
Prymnesioph vcean 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 Ca 2+-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 [Ca 2 +]e values (typically ...... 10- 7_10- 5 :\1) t.hat. permit.s Ca 2 +-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 Ca 2+ concentrations (10- 2-10- 1
M). It may be that the absence of vesicular Ca 2 + stores in these very thin motile appendages means
that an especially steep Ca 2 + 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 Ca 2 + 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 10 4 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:
10 4 N m- 2
Vorticella
Garchesium
4 x 10 4 - 8 x 10 4 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 10 4 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
enclosed by a rectangle measuring 2 cm x 4 cm. Taking 1 cm
1 x 10- 13 m 2.
+-+
1.1 x 10- 7m, it follows that 8 cm 2
The force exerted is thus (5 x 10 4 N m- 2)(1 x 10- 13 m 2)
=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 10 3
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
k
o)
=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
k
= 2~2 In 2 =0.31 ms- 1
The curve in Fig. 3 begins somewhat above 150 J.Lm.
(11)
Let x denote the length in excess of 150 J.Lm.
Then taking the correspondence 1.1 em...... 50 J.Lm, we have
14
0.2 -- 50
x => x -- 9. 1 ,.."m .
(12)
Since this is but an estimate, I round to 10 J.Lm, whence
Lo = 160 J.Lm.
(13)
L(t) = 160 exp( - 0.3lt).
(14)
Therefore
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),
_ (2.6x 10- 3 m s-I)(J.72x lO- tJ 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,
2 x 10
s
between
and
(2 x 10- 1 m s-I)(~ 5 x 10- 4 m)
10-6 m2s- 1
_
=
I.
Accordingly, Re must lie
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
R e>
(10-4 m s-I)(10-4 m) _ 10- 2
-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
(m s-l)
Echinosphaerium
akamae
2.6 x 10- 3
Actinocoryne
contractilis
'"" (2 x 10- 2_
2 x 10- 1)
Heterophrys marina
and
Ciliophrys marina
Actinophrys sol
I
(m)
Re
1.72 x 10- 4
0.4
'" (3 x 10- 5_
3 x 10-4 )
0.7-7
REFERENCE
Suzaki et al.
(1992, p. 432)
Febvre-Chevalier
(1981); FebvreChevalier and Febvre
(1992)
'"" 2.2 x 10- 3
4.5 x 10- 5
0.1
Davidson (1975)
> 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.
e
Happel and Brenner (1983, p. 3) seem to suggest that the creeping motion (Stokes) equations can
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
A
run~
swimming
A duck flying
A luge
:l.t
~t
~
10 m sOl
s~e
the
300.000,000.
speed
30,000,000
20 m s"t
dr~gonfly going
A copepod in
~t
300,000
7 m SOl
30,000
pulse of 20 em SOl
300
Flight of the sm:z.llest flying insectS
30
An
A.
invertebr~te 12.CY~,
0.3 mm long,
moving at 1 mm sOl
0.3
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.
Body
length
Name
Group
(L)~
Swimmingt
velocity
body
~S-I lengths
S-I
Uronema
Tetrahymena
Didinium
Paramecium
Veliger (Aplysia)
Blepharisma
Opalina
Spirostomum
Convoluta
Pleurobrachia
Protozoa
Protozoa
Protozoa
Protozoa
Mollusc larva
Protozoa
Protozoa
Protozoa
Flatworm
Ctenophore
2S
70
130
210
11 SO
480
12S0
1000
ISO
?
350
600
50
1000
600
400
1000
2000
15000
7SOOO
46
7
10
5
?
2
0'12
1
0'3
5
Rb
Number
of body
cilia
(n)
Cilia
length
(l)1Jl1I
10- 2
10- 2
10- 1
10- 1
200·
S
Soo·
17S0
SOOO
7
10- 1
10- 2
1
1
10 3
200
7000·
lOS
lOS.
7 x 106
200(108)
12·S
12
56t
7·5
IS
12
7·S
SOOt
-
Rc
nl
Metachronism
10- 3
10- 2
10- 2
10-%
10- 1
10-%
10- 2
10- 2
10-%
10
10 3
?
10 3
?
2·2 x 10· Dexioplectic
6 x 10· Dexioplectic
1-1 X 10· Laeoplectic
S x 10·
?
1·5 X 10 6 Symplectic
1·2 x 106
?
5 x 10 7 Laeoplectic
lOS (l010) Antiplectic
3'S
X
·Excludes compound cilia; tindicates compound cilia; tfigures for swimming velocity are "maximal" values, particularly for Pleura-
e
brachia.
R and R c are Reynolds numbers based on cell body length 'and cilium length, respectively.
b
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
I
h
The formula in question is appropriate for a
«: hand
C _
S -
L
2~ ~1:
2irf.l
In (Lao)-1.75
(Brennen and Winet, 1977, Table 2, p. 350), where f.l
(1)
= 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
F tot = Cs UL o ,
(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:
Here Lo = 50 /-lm, a = 0.05 Jim, and U = 2.5 x 10- 3 m s-I (Davidson, 1975).
560h, which likely satisfies the condition a ~ h demanded by equation (I ).)
=3xlO- 6 N m- I
= 3 pN /-lm- 1
F tot
= (3 pN /-lm- I )(50 pm) = 150 pN
P s =(3xlO- 6 N m- 1)(2.5xIO- 3 m $-1)
= 7.5 x 10- 9 W m- 1
= 7.5 fW
J.lm- I
P tot = (7.5 f\N J.lm- 1)(50 J.lm)
= 3.8 x 10 2 fW
(Observe that a =
Hence
38
Echinosphaerium akamae:
From Suzaki et al. (1992a, pp. 431-432), I take Lo = 200 11m, U = 2.6 x 10- 3 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
P s = (6 x 10- 6 N m- l )(2.fi x 10- 3 m
5-
1)
= 2 x 10- 8 W m- I
P tot
= (1 x 10- 9 N)(2.fi x 10- 3 m 5- 1)
= 1 x 10- 12 W
= 1 x 10 3 fW
± 0.9) x
10 3 pN.
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
r1 81/J
80
= - U,
~1/Jr
v = 0
at 0 = 0
(4)
- 0
0
r1 81/J
80 = 0, 81/J
8r at = Q.
41
Taylor (1960) and Moffatt (1964) found that the problem has a similarity solution of the form
1/J(r,B) =
Y
2.1. _
'f/ -
1/Jrr
l'
f(B).
1
(5)
1
+ r 1/J r +"2
1/JBB'
l'
(6)
Hence, inserting (5) into (6),
y27j) =
Vf + <10d
1'\
2
;)
whence
which reduces to
(7)
(Note the singularity at
l'
= 0.)
Consider
(8)
The characteristic eq uatioll of (8) is
or
Hence m
= ± i are roots of multiplicity two,
whence
42
Taylor (1960) and Moffat (1964) found that the problem has
ljJ(r,B)
= r frO).
it
similarity solution of the form
(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 +2mor
Hence m
= ± i are
roots of multiplicity two, whence
+1=
0
44
[(0) = AcosO + BsinO + COcosO
+ DOsinO,
['(0) = - AsinO + BcosO + C(cosO - OsinO)
where' denotes
(9)
+ D(sinO + OcosO),
(10)
l(r
Now (4) implies that
EN
[(0) = &'
(11 )
,
f(O)
.
= r1 EN.
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
•
['(a)
f(O) = 0 :::} A = 0
(13)
['(0) = - U :::} B + C = - U
(14)
[(0) = 0 :::} Bsina + Cocoso + Dosino = 0
(15)
= 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
- aSIllO'
si nO' + O'coSO'
acosa - sina
as III a
C
•
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,
D=
U(a-sina cosO')
.'
0'2 _ sin 2 O'
(22)
46
Recalling that t/J = rf(O), and inserting the expressions (13), (19), (21), (22) for A, B, C, D into (9)
yields
t/J=
U~
[-a 2 sinO+sin 2 a(OcosO)+(a-sinacosa)OsinO].
2
2
0' - SIn 0'
In the case of normal incidence,
= f.
0'
t/J =
(23)
Thus, the streamlines are given by
--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-<iimensional Row in a cotner due to ODe rigid plme
sli~ aD anomer (arbitrary =its for !f).
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 1 0
= 0 = JJ {r %r
C/)+¥ ~~r}o = 0
=
o.
From (3) and (23),
Uo
= - ~t/J =
vr
~
2 2 [ - 0'2
0' - SIn 0'
sinO + sin 2 0' (0 cosO) +
(0' -
sinO' cosa)O sinO]
(0' -
sinO' cosa)O sinO].
whence
r
g(:0) = r(a
ar\
2
-
U.
2
SIn 0')
[ - 0'2
sinO + sin2 0' (0 cosO) +
(25)
47
Therefore,
(26)
Also from (3) and (23},
ur
w
=}88 O· =
a
U.
2
-Sill
2
a
[-a 2 cosO+sin 2 a(cosO-OsinO)+(a-sinacosa)(sinO+OcosO)]
whence
u
} 88 Or =
2 U. 2
[a 2 sinO - sin 2 a (2sinO + 0 cosO)
r(a -sm a)
+ (a -
sina cosa)(2cosO - 0 sinO)].
Accordingly,
8pr I
u
-
_
~O 0 - 0 -
2U
(
.
2
• 2
a-smacosa).
r(a -sm a)
(27)
Thus, by substitution of (26) and (27) in (25),
T rO
. 2
I 0 -- 0 = r(a 2 2~U
-sm a)
(.
)
a - sma cosa .
(28)
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,
liD
=
4~U. 2 (
2
a -.
sma cosa ) .
r(a -sm a)
The criterion for the self-consistency of the assumption of negligible inertia is
(29)
48
- rU
Re-zr
where Re denotes the Reynolds number, and
1/
~
1,
(30)
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
1/
(31)
r~U·
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 m 2 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.
•
length of the axopodium, 50 j1.m, is
k(400
Note that the initial
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
Echino&phaerium
akamae
FORCE
POWER
(pN)
(fW)
4 X 103
103
._-----
103
SPECIFIC POWER
(W kg-I)
--------
HOW VALUES
REMARKS AND REFERENCES
OBTAINED
Glass needle deflection
10°
SBT
Suzaki et al. (1992)
Specific power based on est. mass of
entire axopodium; de Mestre and
Russel (1975); Brennen and Winet
(1977)
103
Heterophrys
102 •
103
5 X 101_1 X 10 2 t
5 X 10°_2 X 10 2
(1.1-7.5) X 102 •
SBT
Specific power based on est. mass
of X-body; idem
.Cak./measurement by
Davidson; tCak. by
B.E.D. from Davidson
cine data, Stokes's law
applied to transport
marina
e
Specific power based on est. mass
of entire axopodium; Davidson
(1975)
of attached carbon
sphere
1.5 X 10 2
3.8 X 10 2
1.8 X 104
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
4 X 10°
1 X 10 1
_.. _-----_ ...
---- ..... ----
SPT,
a
SPT, a
= 23
1r
idem
=
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.
colpoda, but I have a value for the smaller C. campy/um.
generally somewhat larger than C. campy/urn:
length
20
J lack the heat of combustion of C.
[According to Blick (1972), C. co/poda is
C. colpoda length = 100-150 JIm;
C. campy/um
= 50-120 JIm.] Laybourn-Parry (1984, p. 140) gives for C. campy/um a heat of combustion of
± 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 10 4 Jlm 3 for C. campy/urn. I shall assume a cell volume of 6 x 10 4 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),
then for one C. campylum cell
Dry mass
~ 0.2(6 x 104 Jlm 3 )(1.0 gjl0 12 J.Lm 3 )
= 1 x 10- 8 g
(1)
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)(10 3 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
Time for assimilation
(3)
4 x 10- 5 J
= 3 X 10 6 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 10 4 pm 3 .
53
The same typical A. sol has
~
42 axopodia, each of length Lo
~
100 11m and basal diameter d A = 1.25
pm. Thus the aggregate volume of the axopodia is
VA = 42 X /2 il"Lod A
2
(5)
= ~~( 100 Ilm)(1.25)2
= 1.8 x 103 pm 3.
The total volume of one A. sol
IS
Vtot=VB+VA
(6)
= (4.8 x 10 4 + 1.8 x 10 3 )
/lm 3
=5.0 x 104 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 (pm 3 ) 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 (4.76.2) x 10 5 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.
Actinopoda.)
diameter of
(Like heliozoa, radiolaria are members of the
Anderson et al. (1984, p. 206) note that the species they studied had a cell body
~
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 10 3 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
minimum of 25 cells) is 818
j.lm
3. 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
10 3.2 neal
= 1.7 x 10 3 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 10 7 fW.
Since the cell body of Spongodrymus is :::: 800 jJm in diameter, its volume is approximately
VB = i7l"(800 jJm)3
- ?_. 7 x
10 8 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
Spiro.ftomum
VOLUME
RESPIRATORY POWER/POWER UPTAKE FROM PREY
(jJm3 )
(fW)
1.2 x 10 7
6.7 X 10 7
1.25 X 108
3.3 X 10
3x 108
9.7X 10
amhiguum
(heterotrich)
Roulina
leei
(foraminiferan)
Spongodrymu.f
(radiolarian)
8
7
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 10 3 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 10100 = 1 x 10-8 % of the energy taken in by Spongodrymus and
3
-13
1.4 x 10- J
x 10 5
100 = 5 x 10- 7 % of the energy obtained by A. sol in the feeding bouts recounted above.
4 x 10- J
(2
(2
J)
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
( 10- 6
10- W
W) 100 = 10-4%.
Note that even if all ::::: 42 axopodia contract at once, the fraction is only about 10- 3 %.
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!
(61ta11v) X v
power
where
11
viscosity of water (lO-2 g/ cm sec)
a
radius of cell
v
= velocity of cell
If the cd! h::s a r.o.dius of 1 J.UI1 and is trd.vding at 10 J.UIl!sec, che power consumed is 2 X lOo11 e.rgsIsec (2 X 1O-l8 J/sec).
What is this power requirement in tc:III1S of ATP molecules, che principal currency of energy in the cell? Hydrolysis of one gram mole of ATP releases
about 40KJ of useful. energy; hydrolysis of a single ATP molecule, abOUt 10'1' J.
The cd! in the above calculation chert:.:are requires che"hydrolysis of at le::sc 20
molecules of ATP every second in order to maintain its speed of 10 ~sec. Ifwe
::ssume a low efficiency of a few' percenc.ge points., chen che ATP hydrolysis due
to swimming might be 1000 molecules per second. Since che mecboiic me of a.
typica.l cd! is around 10' ATP molecules per second, ho~, ~g is noe
a ma.jor energy case.
'.,
Hence, Bray's fictive coccus devotes only 10- 2% of its metabolic power to locomotion.
It is useful to
58
An analogous estimate was made by Pedley and Kessler (1902, p. 318):
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 of 3%,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 input energy. Estimates for spermatozoan swimming power range from
2 x 10- 15 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.
ENERGY
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
Cell radius
~
43% of extended length (p. 599)
=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
(Jlrn)
TYPICAL SPEED
(J.lrn s·l)
TYPICAL TIME
(rns)
ENERGY
(J)
Spiro8tomum
ambiguum
855
2 X 10 5
-4
2.13 X 10. 10
Vorticella
conl1allaria
80
2.3 X 10 4
-3
6.9 X 10- 13
172
2.6 X 103
65
2 x 10- 13
:
Echino8phaerium
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 (FebvreChevalier, 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,
polymerization/ depolymerization.
others
on
cell
movement
and
filament
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
contraction.
> 1 basic mechanism of
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 colchicinesensitivity, 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?
achieve steady speed?
XI. INTERNAL HYDRODYNAMICS
What is the characteristic time to
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.
Echinosphaerium akamae.
(1980)
observations
of spontaneous
axopodial
contraction
lD
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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[Discussion and EM images of MT triads supporting the tentacles.]