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.