Proc. Natl Acad. Sci. USA Vol. 78, No. 9, pp. 5613-5617, September 1981 Cell Biology

Microfilament or microtubule assembly or disassembly against a force (actin/tubulin/mitosis/cell shape change/siclde cellhemoglobin) TERRELL L. HILL Laboratory of Molecular Biology, National Institute ofArthritis, Metabolism, and Digestive Diseases, National Institutes of Health, Bethesda, Maryland 20205 Contributed by Terrell L. Hill, May 5, 1981 ABSTRACT Microtubules (tubulin) or bundles of microfila- An idealized aspect of the treatment to be given is that the ments (actin) are thought to cause movement, in some instances, pool ofsubunits or monomers used for the polymerization pro- by disassembly or assembly ofsubunits. Possible examples are the cess is assumed, for concreteness, to be a dilute solution. In pulling ofa chromosome toward a pole in mitosis (anaphase) or the some cases the actual pool may be rather concentrated, or there deformation ofa cell membrane to change the shape ofa cell. This may even-be an intermediate bound state for the subunits (5). paper examines the relevant elementary bioenergetic considera- Such conditions would not alter the formal thermodynamics tions when assembly or disassembly ofan aggregate occurs against much but could change the kinetics significantly. a resisting force. The problem is considered, in the fist section, without NTPase activity. Siclde cell hemoglobin aggregation in vivo is an example. In the second section, the tubulin GTPase and ASSEMBLY-DISASSEMBLY AGAINST actin ATPase activities are included in the analysis. A FORCE, WT1HOUT NTP Many ofthe essential features ofinterest here are not concerned In aprevious companion paper (1), bioenergetic principles were with the NTPase activity ofthe aggregating subunits. Therefore considered which related to enzyme translocation on DNA and we consider first, in this section, the much simpler case of as- to the treadmillingofone-dimensional aggregates ofDNA-bind- sembly-disassembly without such activity. ing proteins that are bound adjacent to a replicating fork. Here Fig. 1 illustrates very schematically the kinds ofsystems we we examine possible motion that results from the lengthening consider. Fig. IA represents a microtubule (of13 strands) or an or shortening ofmicrofilaments (actin) or microtubules (tubulin). actin microfilament (of2 strands), orabundle ofmicrofilaments, Polymeric actin (2) and microtubules (3) are known to be in- under a total compressing force F, which we arbitrarily give a volved in a number of examples of motility. In most of these negative sign. Fig. lB shows a microtubule under an extending cases the necessary force generation presumably arises from force F (positive). In both cases there are binding (from solution) actin-myosin or from microtubule-dynein interactions, or vari- and release transitions of individual subunits at one end of the ations on these. However, there have been suggestions (4) and polymer, which can result in length changes in the polymer. there is evidence (2, 5) that assembly or disassembly of micro- These length changes have to contend with the force, F. The tubules or ofactin microfilaments might themselves be directly other end of the polymer is assumed (to simplify equations) to responsible for some kinds ofmotility. Examples are: the pulling have an essentially permanent and nondynamic attachment of a chromosome toward a pole in mitosis (anaphase) by disas- (13). In Fig. LB, we assume that a microtubule with 13 strands sembly, at the pole, of microtubules that are attached to the can still maintain attachment to some other nonrigid cellular kinetochore (4, 6); and protrusion ofa cell membrane as a result structure at the top, under an extending force, even if several of actin polymerization (5, 7). The former example is likely to subunits out ofa ring of 13 are missing (owing to the transitions be of more importance as a rate-determining process (6) than referred to). The same assumption is plausible for a bundle of as a source offree energy to move the chromosome (4), but this actin microfilaments, though not for a single microfilament (two is not definite yet. There are examples of the latter type that strands). We do not propose an explicit mechanism for the se- almost certainly involve actin only, and not myosin (2, 3). The quence ofsubunit departures, and possible shifts in subunit lat- in vivo aggregation ofsickle cell hemoglobin is also an example tice positions, that would allow simultaneous shortening and of cell membrane distortion. maintenance of contact with the other structure. The object of this paper is to outline elementary theoretical Fig. IC illustrates the special case ofFig. 1A ofmost interest: principles that are pertinent for systems in which either disas- the concentration c of subunits (also called monomers) in so- sembly or assembly of a filamentous aggregate can do work lution is large enough so that there is net aggregation (growth) against a resisting force such as a chromosome or a cell mem- at either end ofa microfilament bundle, despite the compress- brane (i.e., a deformable surface), respectively. ing force F < 0 (from a deformable membrane) that resists the A complication. that will be included in the second section growth. Similarly, Fig. ID is the special case ofFig. lB ofmost is the fact that microfilament and microtubule assembly or interest: despite the extending force F > 0 from a chromosome disassembly requires hydrolysis of ATP or GTP, respectively that must be dragged through the solution, the microtubule (2, 8). That is, in these processes, actin is an ATPase and tubulin shortens from a net loss ofsubunits at one end. We give a single is a GTPase. This NTPase activity leads, in solution, to steady- treatment, below, that applies to both of these cases. state treadmilling, also called head-to-tail polymerization, as is The second-order "on" rate constant for attaching subunits well known (9-13). at the "dynamic" end, from solution, is a; the first-order "off" rate constant is a'. When F = 0, these rate constants are des- The publication costs ofthis article were defrayed in part by page charge ignated by a0 and a'; a and a' depend on F. Even at F = 0, payment. This article must therefore be hereby marked "advertise- a0 and a' presumably have somewhat different values than for ment" in accordance with 18 U. S. C. §1734 solely to indicate this fiact the same polymer end when it is free and uninhibited in so- 5613 Downloaded by guest on September 26, 2021 L56014 Cell Biology: Hill Proc. NatL Acad. Sdc. USA 78 (1981)

f -1

1/2 fM/F 0 F

I Membrane Pole FIG. 2. Example of possible F dependence-of kinetic parameter f. See text for further details.

~~~D - 1/F for large positive F. The latter is meant to produce the (F 0) diflusion-controlled limit for a, which ought to be larger than a0 because of interference in the attachment process by the neighboring structure (Fig. 1B) when F = 0. The subunit thermodynamic force, tending to produce ag- gregation, is Apu - - . This is also equal to Rfln(c/c°). At =-IF (Eq. 2), or Au + IF = 0. This sum FIG. 1. (A) Exchange of subunits between a solution and one end equilibrium, AI of an aggregated filament or tubule that is under a compressing force, is the total-thermodynamic force, for aggregation, including the F < 0. (B) Same for an extending force, F > 0. (C) Special case of A resistance term. The total force has the value zero at equilib- in which net addition of subunits causes lengthening of an aggregate rium; at arbitrary c, it is equal to RTln(c/c.). (e.g., a bundle of microfilaments), thus pushingback a resisting mem- Fig. 3 summarizes six special cases that have to be consid- brane. (D) Special case of B in which net loss of subunits in a micro- ered, depending on the values ofc and F. The ordinate in the tubule causes shortening and the-pulling of a chromosome toward a figure-is In c; the abscissa is F. The heavy lune shows the rela- mitotic pole, against a resisting frictional force. tionship between ln c and F at equilibrium. Explicitly, from lution. A "capped" filament,- for example, has F = 0. Eqs. 2-and 4, The chemical potential of subunits in solution, at concentra- ln ce = hI ce - (IF/RT). [8] tion c, is For all points in the plane above this line, we-have c > c6, AA Ap + RTln c, [1] + IF > 0, and net aggregation (lengthening) occurs. For all where R is the gas constant and T is the absolute temperature. points below this line, C < C., Aa + IF <0, and disaggregation (shortening) occurs. To the right ofthe ordinate, IF >40; to its In the polymer, the subunit chemical potential is j., which, for simplicity, we take to be independent of F or of length. That left, IF < 0. The horizontal line that crosses the heavy line cor- to C = (i.e., the subunit force ApL is zero on this is, the polymer is considered incompressible and we neglect any responds cP, thermodynamic end effects for finite polymers. The average line); Above the horizontal line, ApA> 0; below- the line, Ap length contributed to the. polymer by one- subunit is 1. In a <0. microtubule, for example, I = 80 A/13 = 6.2 A. When one sub- The cases a and d in Fig. 3 are relatively uninteresting be- unit is removed from the solution and added to the polymer,- cause the two subforces (A4 and IF) have the same sign as the the Gibbs free energy changes are -,a in the solution, ye in totalforce. But in cases b, c, e, and f, the subforce with the same the polymer, and -IF in the "resistance" (membrane, chro- sign as the total force is larger in magnitude than the other which has the opposite sign. Consequently, there is mosome, etc.). Atequilibrium, C Ce, ace = a' (detailed balance), subforce, and free energy transduction in these cases, as we now explain. For example, in case c (corresponding to Fig. IC), the sub- .Lo-IF = u, = ? + RTln ce [2] = ts + RTln(a'/a).. [3] These equations show how c, and a'/a depend on F. When F = ° C-C, = a ,.and-- y0 = ju4 + R71n co = ,4 + RTln(aG/ao). [4] The separate dependences of a and a' on F can be written in a formal way as a = aoeflFRT [5] a' = aoe(f- 1)lF/RT [6] wherefis a dimensionless parameter that expresses the "split" of IF between the on and off rate constants-. Ofcourse, fdrops out of the quotient-: a/a' = (adao)el/RT. [7] The parameterfitselfis afunction ofF. That is; the split would FIG. 3. The heavy line shows the relationship between hn c and a be expected to be different under compression (F < 0)-and un- mechanical force F, at aggregation equilibrium, where c is the mono- der extension (F > 0). Fig. 2 gives a hypothetical illustration, mer concentration. The special cases a-f refer to nonequilibrium sit- in which f = 1'2 (symmetrical split) for large negative F andf uations (off of the heavy line), discussed in the text. Downloaded by guest on September 26, 2021 Cell Biology: Hill Proc. Nad Acad. Sci. USA 78 (1981) 5615 unit concentration is large enough so that assembly and length- system in view of the fact that we have not yet included it in ening occur despite the compressing force (F < 0) which op- the model. The aggregation of sickle cell hemoglobin does not poses lengthening. Part ofthe subunit free energy decrease Ap involve NTPase activity. (per subunit added to the polymer) is used to do an amount of work -IF against the membrane (Fig. 1C). The efficiency ofthe ASSEMBLY-DISASSEMBLY AGAINST A FORCE, transduction is 7 = (-lF)/AM. The source of the free energy INCLUDING NTP used is the subunit pool at high enough chemical potential /,u. We now reconsider systems of the type shown in Fig. 1, and Ifc is decreased (keeping F constant) enough to cross the heavy include the NTPase activity that actin and tubulin actually show line in Fig. 3, case c passes into case b. Here AM and IF have when they polymerize. The approach that we use is very similar the same signs as in case c, but there is role reversal: the com- to that in ref. 12, where the bioenergetics ofsteady-state tread- pressive force, the free energy. source in this case, is large milling was examined. Here, only one end of the polymer can enough to drive subunits out of the polymer even though AM gain or lose monomers, the other end beingblocked, and steady > 0 (i.e., c > ct). The efficiency is q = AA/(-iF). state (constant length) is ofonly incidental interest. Treadmill- If a free polymer, growing at c > co and F = 0, encounters ing (head-to-tail polymerization) is not possible. We take over a rigid barrier, F will quickly decrease at constant c until growth the notation ofref. 12 except that Apr (instead ofX) is the NTP stops (c = ce line). free energy of hydrolysis and c (instead of c1) is the monomer Case fin Fig. 3 corresponds to Fig. ID. Here the pool con- concentration in solution. Explicitly, Apr = MT- AD - AP centration c is sufficiently low so that disassembly and short- AT, ATP; D, ADP; P. Pj). ening occur despite the extending force F > 0. Part ofthe sub- Fig. 4A shows a possible NTPase cycle (12, 14), where A rep- unit free energy decrease -AM is used to do work, in the resents a monomer, AT a monomer with NTP bound, etc. There shortening process, against the resisting force F. The efficiency is only a partial NTPase cycle for monomers in solution and a of transduction is q = IF/(-AM). Of course, in the example in complementary partial NTPase cycle for a terminal monomer Fig. 1D, the work against F is not stored (as it would be if, say, ofthe polymer (a nonterminal monomer in the polymer is frozen a weight were lifted) but rather it is dissipated as heat in the in state AD). The two partial cycles, however, form a complete viscous medium through which the chromosome moves. Also, cycle, as indicated in Fig. 4A. We make the usual assumption in this example, F itself would be proportional to the rate of (11) that the two boxed species in Fig. 4A are dominant. The shortening (see below). Incidentally, a chromosome, in ana- monomer concentration c thus refers to AT. With this simpli- phase, has a small velocity ofonly about 1 pum min-1 (6) or 0.2 fication, we can replace the six-state cycle in Fig. 4A by the two- A Insec1 (the velocity in muscle contraction is of order 10 A state cycle in Fig. 4B. The latter figure includes the rate con- msec-). stant notation; a, and a2 are second-order constants, whereas Ifc is increased sufficiently, in case f, holding F constant, case a2 and a-, are first-order constants. These rate constants refer ftransforms into case e. Again AM and IF retain the same signs, to whichever end ofthe polymer (the two ends are intrinsically and there is role reversal: the extending force is now large different) is able to exchange monomers with the solution. The enough to induce aggregation of subunits even though Ap. < rate constants a, and a2 predominate in the kinetics; a-, and 0 (i.e., c < c°). The transduction efficiency here is v, = (-,u)/ a2 are relatively small (but they are needed for thermodynamic IF. purposes). We define the flux Jm as the net rate of adding monomers The chemical potential ofa monomer (AD) in the polymer is to the polymer: Jm = ac - a'. The velocity of lengthening of designated by /.AD, a constant. The monomer (AT) chemical the polymer is then v = JmI. On using Eqs. 5 and 6, Jm becomes potential in the solution is written 0 MuAT = MAT + Rn c. [11] Jm = aI e(f )lF/RT [e(+lF)/RT - 1]. [9] Then if we consider a hypothetical equilibrium, including the At equilibrium, both flux and total force are zero: Jm = 0 and from transitions only, we AM + IF = 0. Otherwise, Im and AM + IF have the same sign. mechanical force F. resulting a,, a1 Near equilibrium, [ ] in Eq. 9 is replaced by (AM + lF)/RT. obtain the analogue of Eq. 3: In the absence ofan outside mechanical force (F = 0), the rate MAD + AP - IF = MAT + RTln(a-1/a1). [12] of aggregation Im is simply equal to a'(eAs/RT - 1). The re- maining thermodynamic force here is AM. On rearrangement, The rate offree energy dissipation in this system is Rf1n(a/a.-1) = MAoT - (>AD + MP - IF). [13] TdiS/dt = (ac - a')(A, - MO + IF) = Jm(AM + IF) - 0. [10] Similarly, for the a2, a-2 pair, we find Recall and AM + IF always have the same sign and that, thatjm In Soklion In Solution in the transduction cases b, c, e, and fin Fig. 3, one ofAM and A IF is positive and the other is negative. a In Eq. 9, fis some function ofF, as already mentioned. One Detach I II 02 (l-2AT a)2 would expect each particular system to have its ownfRF). Also, IXIkTIAttach in the chromosome or any similar case (Fig. ID), Jm and AM + IF are negative and F = -fJml, where F > 0 and (3 is the AD PolTe frictional coefficient. This connection between F and Jm con- On Polymer End AD verts Eq. 9, in such a case, into an implicit equation in Jm. (A) In summary: In the cases c (Fig. IC) and f(Fig. ID) ofprac- AD tical interest, there is no difficulty, in principle, with sugges- (B) tions that subunit respectively, aggregation or disaggregation, FIG. 4. (A) NTPase cycle in the attachment-detachment of sub- can do work against an opposing mechanical force. The driving units at one end of an aggregate. A represents one subunit. The boxed free energy for this work is a subunit chemical potential differ- states dominate in the cycle. (B) Rate constant notation for two-state ence; NTPase activity is obviously not a required feature ofthe cycle, using "boxed' states only. Downloaded by guest on September 26, 2021 5616 Cell Biology: Hill Proc. NatL Acad. Sci. USA 78 (1981)

RT1n(aJa_) = (FAD + AP - IF) - VAT - ApqT). [14] Inc Both a/a-1 and aJa-2 depend on F. The successive basic free energy levels (12, 15) in a cycle, starting from AT in Fig. 4B, are /AT, PAD + P - IF, and FPAT - Ap (Eqs. 13 and 14). The middle level depends on Fbut the overall cycle thermodynamic force does not: on adding Eqs. 13 and 14, we have ala2/a-la_2~= eAT/RT. [15] I(c) The total thermodynamic force is Apr, of order 12 kcal mol'1 A_ + IF O.AjI+ > 0 above c = c%,but a categorical state- AU3 /AAT + R71 c- Apr. [18] ment about Aju+ cannot be made below c4. ForF < 0, Aju < 0 below c = c%, but a categorical statement about Au- cannot be made above These will be needed below. In a complete cycle, Al- 3 = ct. See text for further details. Ap. The net transition fluxes (15), in the main cycle direction (Fig. 4B), are Jm = (a2 + a.l)[c/c,) - 1]. [25] - J, ai - a-,, 12-a2 -a2c. [19] The quantity (c/c<,) 1 is an apparent thermodynamic force defined about the steady state; but c,, is a kinetic property (Eq. The net rate of adding monomers to the polymer is then 21), so that this is not a legitimate thermodynamic force. Sub- stitution ofEqs. 21-23 in Eq. 25 givesJ. as an explicit function Jm = 11 -12 = (a1 + a-2)C - (a2 + a..4 a1c- a2. [20] of F. At steady state (constant length), Ji = J2, Jm = 0, and c c,,, The expression forthe rate offree energy dissipation provides where the most insight into the nature of this system. We start with the "transition" relationship (15) c, = (a2 + a-,)/(a, + a_) a2/al. [21] TdiS/dt = JI(Al - A) + J2(A2 - A3) 0, [26] When c > ca, J. > 0 (the polymer lengthens); when c < c., Jm < 0 (polymer shortens). Thus the condition c = c,, for Jm where the detailed expressions are given in Eqs. 16-19. This = 0 replaces c = c,, in the previous section (Fig. 3). Note that single formal equation applies to all cases. But, for conceptual C. aJal, compared to c, = a'/a above. In c,,, a2 and a, advantage, we rearrange the terms here in two different ways belong to different transition pairs (Fig. 4B). The equilibrium depending on whether c > c,, (polymer lengthens) or c < c,,, concentrations for the individual transition pairs would be ce ) (polymer shortens). > we have = - SO that > = a-ja, (very small) and ce2)= a2a-2 (very large); c,, ad When c coo also Jm Jh J2> 0, J1 al is intermediate, of order 1 ,AM (10). 12. The two transition fluxes J, and J2 are represented sche- The rate constants in the above expressions depend on F. matically in Fig. 6A. Because J1 >12, J1 is subdivided into two Corresponding to Eqs. 5 and 6, we write parts (arrows): an amountJ1 to match the other transition flux; and the excess J,- 12.2 now represents a complete cycle flux a a 1 a, lefIlFIRT,a a- e(fi-l)IFIRT [22] (Fig. 6A); in acomplete cycle one molecule ofNTP is hydrolyzed -e~,a v/ R but the polymer remains unchanged. We therefore define the 0=a-efiF/RT a e(l-f2)IF/RT [2] a2 2e , a2 a.2 [3 NTP flux as 12(+)-l (the + refers to the lengthening case, c where f1 and f2 are parameters that themselves depend on F > c,,); the associated complete cycle thermodynamic force is (see, e.g., Fig. 2), and acj, etc., are the rate constants at F = Apr. The excess flux J, - 12 = Jm ("on") is the net rate of ad- 0. Substitution ofEqs. 22 and 23 in Eq. 21 gives c0, as an explicit dition ofmonomers, occurring via a1, a-, transitions (Fig. 6A). function ofF. At F = 0, c,, c9 . Ifc,,(F) is plotted as ln c,,(F), We break down the total thermodynamic force g, - p (Eqs. the curve (Fig. 5) would in general not be linear (compare Fig. 16 and 17), associated with this excess a1, a-, flux, into the 3). But ifa-, and a2 can be neglected (one-way cycle), we have subunit thermodynamic force and the "resistance" force: in c. = in c! - [(f1 + f2)IF/RT], [24] AT (Sokubon) AT (Souton) which is linear iff1 + f2 is constant. We also obtain a linear re- J J 3)JI-2 J2 J2-Jl JI )JI lation, as in Eq. 8, for the two-way cycle, iff' + f2 = 1 (the A (PoiwM) AD (PoIMr) special case fj + f2 = 1 simulates an equilibrium). If the tran- C > CM c

= + = C Al, 2 An+ IF [27] TdaS/dt Jm(Ap.u + IF) + j1() AFT (C c). [3] AA+ = (pAT + RTln c) (AAD + AP)- This equation is the companion of Eq. 28. In this case, Jm and A/& + IF are both negative, while j I-) and Aprare both pos- Here AA,+ is analogous to Ap in the previous section. With itive. Onthecurvec = c.,,Jm = 0but AA_ + IF 0, as in Fig. iD. Some subunit free energy is used to do Compared to the analogous Eq. 10, there is a new term here work against the resisting force F. The efficiency of the trans- representing the dissipation of NTP free energy. When c = c., duction is the first flux-force on the of 28 out expression right Eq. drops 77 = (-JjlF/[(-Jj)(-At..) + J () AFTr] (c _ C,). [34] (because Jm = 0), but J(+) AprU remains. In this case (c = c.o, steady state), NTP is being used but the polymer does not Again the NTPase activity reduces the efficiency. Without the change; no work is being done by the system. Incidentally, c NTPase term, the efficiency would be lF/(-Au-). = c,, is not, as might be expected, the value of c at which TdiS/ Quite aside from the wasteful NTPase dissipation terms in dt is minimized (1). Eqs. 28 and 33, the thermodynamic forces AAu and -AA- (of Eq. 28 applies on or above the curve in Fig. 5. Case c in this order 6 kcal mol-') seem unnecessarily large for the purpose figure is not very interesting, but case b involves free energy at hand. These forces are related to the ratios a/a¶ll, and aj/ transduction: the filament or polymer lengthens against a com- a%, respectively. But the quantity of practical interest here pressing (negative) force F, as in Fig. 1C. The efficiency of the (rate of assembly or disassembly) is Jm alc - a2, which de- transduction is pends essentially only on the rate constants a, and a2 and not on and a-2. In the previous section NTPase 27 Jm(- F)/(Jm&AP+ + JT) ApF) (C 2 CO)r [29] a-, (no activity), on the other hand, Jm = ac - a' and AA. is related to ao/ao: The efficiency is reduced by the presence of the NTPase term. flux and force depend on the same (inverse) rate constants, Without this term, the efficiency would be -lF/IAp+. In fact, which is more efficient (there is less free energy dissipation). the NTPase activity serves no obvious purpose here (see the On the curve c = c,,, there is a discontinuity between Apt+ preceding section). However, there could, conceivably, be and Ap_ (but Jm = 0). On the other hand, J(+) = J(-) on this some kinetic advantage to the use of the NTP cycle in Fig. 4 curve, because1 = 12. There are, of course, no real discontin- rather than simple assembly-disassembly transitions. uities across c = c,,; we have merely, for conceptual reasons, If a free polymer, growing at c > ca and F = 0, encounters changed our classification scheme (Fig. 6) at this boundary. a rigid barrier, F will quickly decrease at constant c until growth In summary, Eq. 26 has been rearranged in Eqs. 28 and 33 stops (c = c,,. line). If both ends can exchange subunits, tread- in order to separate pure (complete cycle) NTPase activity from milling will occur. the residual monomer activity (addition to or subtraction from In Eq. 10, both Jm and Ap. + IF are equal to zero on the c the polymer). This monomer activity is analogous to that in the = ce line (Fig. 3). In the present steady-state case, Jm = 0 on previous subsection (NTPase absent); the NTPase activity here the c = c,, curve (Fig. 5), but ApA+ + IF > 0. This can be seen is superimposed, and is wasteful. Although inefficient, there as follows. We have, by definition, is no difficulty, in principle, in the use ofassembly or disassem- bly to work against an outside mechanical force, as in Figs. IC + IF = - [30] A,u+(c,) p.,(c,) A2, (cell shape changes) and ID (chromosome movement). where c,,O is given by Eq. 21. If we replace c,,, on the right by I am much indebted to Dr. Marc Kirschner for a helpful discussion. c(l) = a,/a,, the right side becomes equal to zero (equilib- rium). Thus the right side in Eq. 30, as it stands, is positive if 1. Hill, T. L. & Tsuchiya, T. (1981) Proc. Natl Acad. Sci. USA 78, c. > a,/a,, which is the case if a2/a-2> a-,/a, (Eq. 21). 4796-4800. 2. Korn, E. D. (1978) Proc. Natl Acad. Sci. USA 75, 588-599. But this is true in view ofEq. 15. Ofcourse, as c increases above 3. Roberts, K. & Hyams, J. S., eds. (1979) Microtubules (Academic, c = c,,, AAe+ + IF also increases (Eq. 27). Thus, above the c New York). = c., curve, both flux-force products in Eq. 28 have two positive 4. Inou6, S. & Ritter, H. (1975) in Molecules and Cell Movement, factors; both contribute to free energy dissipation. eds. Inoue, S. & Stephens, R. E. (Raven, New York), pp. 3-29. We turn now to the opposite case, c < C,,, in which Jm < 0 5. Tilney, L. G. (1975) in Molecules and Cell Movement, eds. Inou6, shortens) and > In view of Fig. 6B, we define S. & Stephens, R. E. (Raven, New York), pp. 339-386. (polymer J2 J1. 6. McIntosh, J. R. (1979) in Microtubules, eds. Roberts, K. & J(-)-Il (complete cycle), with associated force Apr, while the Hyams, J. S. (Academic, New York), pp. 381-441. excess subunit flux (positive)J2- Ji = -Jm ("off") has the con- 7. Goldman, R. D., Schloss, J. A. & Starger, J. M. (1976) in Cell jugate force A2 - A3 (positive). We subdivide p2 - p3 into sub- Motility, Cold Spring Harbor Conferences on Cell Proliferation, unit and "resistance" terms, as in Eq. 27: eds. Goldman, R. D., Pollard, T. D. & Rosenbaum, J. L. (Cold Spring Harbor Laboratory, Cold Spring Harbor, NY), Vol. 3, p2- .3=-Ap.--F [31] Book A, pp. 217-245. 8. Jacobs, M. (1979) in Microtubules, eds. Roberts, K. & Hyams, J. Ap.u_ =(.AT + RTln c Apr) -(F.AD + AP). S. (Academic, New York), pp. 255-277. 9. Margolis, R. L. & Wilson, L. (1977) Proc. Natl Acad. Sci. USA 74, Again Ap.. corresponds to Ap. in the previous section, but Ap. 3466-3470. is negative for values of c of interest. In fact, 10. Bergen, L. G. & Borisy, G. G. (1980)J. Cell Biol 84, 141-150. 11. Wegner, A. (1976)J. Mol Biol 108, 139-150. Ap.+ Ap. = Apr. [32] 12. Hill, T. L. (1980) Proc. Natl Acad. Sci. USA 77, 4803-4807. 13. Kirschner, M. W. (1980)J. Cell BioL 86, 330-334. Because Ape is of order 12 kcal mol-, Ap.+ and -Ap. might 14. Hill, T. L. (1981) Biophys. J. 33, 353-371. both be oforder 6 kcal mol'. With these definitions ofJ(-) and 15. Hill, T. L. (1977) Free Energy Transduction in Biology (Aca- ApA, Eq. 26 becomes demic, New York). Downloaded by guest on September 26, 2021