RecA Force Generation by Hydrolysis Waves*

We present a simple theory of the dynamics of force generation by RecA during homologous strand exchange and a continuous, deterministic mathematical model of the proposed process. Calculations show that force generation is possible in this model for certain reasonable values of the parameters. We predict the shape of the force-velocity curve for the Holliday junction, which exhibits a distinctive kink at large retarding force, and suggest experiments which should distinguish between the proposed model and other models in the literature.

products of bare ssDNA and dsDNA covered by RecA (17,18). Removal of the RecA from the product is required to allow functioning of the dsDNA and recycling of the RecA (16).
RecA filament dissociation proceeds from the end of the nucleoprotein filament opposite that on which it associates, so depolymerization proceeds 5Ј 3 3Ј (8,9), i.e. disassembly proceeds in the same direction as assembly. Although ATP hydrolysis is not required for polymerization, depolymerization is blocked if the cofactor used in assembly is not hydrolyzable (9), so disassembly is clearly coupled to hydrolysis. ATPfueled depolymerization provides us with important information on the ATPase activity of RecA. ATP is hydrolyzed uniformly throughout the filament (19). 5Ј 3 3Ј disassembly from dsDNA, however, occurs 3 times as rapidly as the average time between hydrolysis events on any given monomer (20). The most natural interpretation is that ATP hydrolysis on dsDNA is a cooperative process (20): the hydrolysis of ATP by one monomer induces the hydrolysis of ATP by its neighbor on the 3Ј side. Cooperative hydrolysis will produce "waves" of hydrolysis propagating along the nucleoprotein filament in the 5Ј 3 3Ј direction (20). Recently rapid disassembly of dsDNA filaments has been monitored on individual nucleoprotein filaments (21).

Strand Exchange
The central function of RecA is to promote strand exchange. Two types of strand exchange reactions are encountered: three-stranded reactions and four-stranded reactions.
Three-stranded Reaction-In the three-stranded reaction, RecA first forms nucleoprotein filaments with ssDNA (22). A region of homology is then found on a dsDNA molecule, most likely by formation of a transient triplex structure (14,22). The RecA-coated ssDNA replaces the identical strand from the original dsDNA, requiring a relative rotation between the two substrates (23). In terms of chemical reactions, the substrates are a RecA-coated ssDNA molecule and a bare dsDNA molecule (1), while the products are a RecA-coated dsDNA molecule and a bare ssDNA molecule (17,18).
Neither the homology search nor a limited (1-2 kilobase pairs) threestranded exchange reaction in homologous regions require hydrolysis of the NTP (24). Hydrolysis of the cofactor is required for exchange of longer homologous regions, for bypass of heterologous inserts, and for the four-stranded reaction (24). It follows from basic thermodynamic principles that, because limited three-stranded exchange reactions proceed spontaneously, the dsDNA/RecA nucleoprotein filament has a lower Gibbs free energy than the ssDNA/RecA nucleoprotein filament. Thus, even though the RecA binding kinetics is more rapid on ssDNA, the double-stranded variant must provide the larger binding energy.
It is this clever "design strategy" that permits spontaneous threestranded strand exchange; the ssDNA filament is a rapidly forming "transition state" that converts more slowly to the dsDNA filament ground state when presented with a homologous dsDNA partner. Hydrolysis plays the role of an auxiliary "engine," which stimulates an already exothermic process, allowing it to overcome roadblocks in the form of heterologies and greatly extending the length of homologous DNA which may be exchanged (24). The final dissociation of RecA from dsDNA now inevitably requires an external free energy source, which is indeed the case as already noted (9).
Four-stranded Reaction-In a four-stranded reaction, RecA forms nucleoprotein filaments with dsDNA. The four-stranded exchange reaction always begins in a three-stranded region. A second dsDNA molecule is again searched for a region of homology. Once this is found, the three-stranded reaction proceeds until the four-stranded region is reached. In the four-stranded region, a strand from one molecule replaces the identical strand from the other, and vice versa (20). The point at which the two strands are being mutually exchanged is known as the Holliday junction. During strand exchange, the Holliday junction moves from one end of the complex to the other, requiring a relative rotation between the two strands along their axes in opposite directions (see Fig.  1) (23).
If the two substrates are parallel to each other as shown in Fig. 1, all three of the motions diagramed are possible (23). Although the motion designated 1 can still accomplish the spooling of DNA strands between the two substrates when they are at a large angle relative to each other, the other two motions cannot. The experimental evidence indicates that the motion labeled 1 is indeed the motion which occurs during the four strand exchange process (23). Some models in the literature will work only if the substrates are parallel to each other (20,25). We will propose below a model which will work both in this case and if the two substrates cross at a large angle.
The substrates in the four-strand exchange reaction are the same as the products: two dsDNA molecules, one of which is coated with RecA. It now follows from basic thermodynamics that the reaction can not proceed spontaneously in a unidirectional manner. The four-stranded exchange reaction indeed always requires ATP hydrolysis (24). In the absence of ATP hydrolysis, a Holliday junction could do no more than aimlessly diffuse along the complex. In practice, the Holliday junction does not even enter the four-stranded region in the absence of ATP hydrolysis.

DNA Binding Sites
There are three binding sites for DNA within the RecA/DNA nucleoprotein filament, denoted by I, II, and III (5,22,26,27). Binding of ssDNA to RecA is to site I (22). Binding to dsDNA is to sites I and II (22), regardless of whether the binding nucleates on dsDNA or is the extension of a filament which began in a single-stranded region of the molecule. During strand exchange reactions, the strand at site I always remains in the RecA helix.
In a three-stranded reaction, the invading strand is bound to site I so the final dsDNA is bound to RecA at sites I and II (17,22). If a three-stranded filament was formed as an intermediate structure during the exchange process, then the strand identical to the invading strand would have occupied site III, from which it was expelled from the filament after the new base pairs were formed in the hybrid molecule.
From the fact that the three-stranded exchange proceeds spontaneously (13,14,24), we conclude that the strongest DNA binding site of RecA is site I, that a weaker binding occurs at site II, and that site III is the weakest binding site. We will denote the binding energies by E I , We take the state of free DNA exterior to the RecA filament and not associated with RecA to have a binding energy of 0. It is not necessarily the case that E III Ͻ 0. So long as (E II ϩ E III ) Ͻ 0, the double-stranded DNA in the three-stranded reaction will be brought into the filament by the net binding energy there. If E III Ͼ 0, it is a repulsive "anti-binding" site which will tend to expel any DNA strand occupying that position. We will discuss this further under "Three-stranded Reaction." In a four-stranded exchange, the displaced strand is assumed to vacate site II when it leaves the nucleoprotein filament at the junction. Site II is then occupied by the incoming strand (22). Site I is always occupied by the strand complimentary to that at site II. Whether site III plays any role is not clear.
Recall that the dsDNA nucleoprotein filament extends DNA by 50% (3). Molecular mechanics experiments on naked DNA show that stretching DNA by that amount requires an amount of work W roughly equal to 10k B T per base pair (28). The free energy gain per base pair obtained in transforming a ssDNA filament into a dsDNA filament is E II ϩ W. It follows that E II Ͼ W, since the reaction proceeds spontaneously.

THEORY
Force Generation Models-How does ATP hydrolysis generate the torque that produces the counter-rotation of the DNA strands during exchange? Mechano-chemical force transduction has been considerably discussed in the biophysical literature. Two classes of force transduction have received attention. Motor proteins like myosin and dynein contain flexible sections. Motion of these sections driven by ATP hydrolysis generates mechanical forces in the 10 pN range (29). Force generation is also possible by the polymerization reaction. Actin and microtubule filaments are able to generate mechanical force by polymerization and depolymerization reactions fueled by ATP or GTP (30). RNA polymerase is a motor protein that also produces mechanical forces in the 20 pN range (31) based on nucleotide polymerization. From our earlier discussion, it would seem very reasonable to expect that RecA generates force by polymerization and/or depolymerization.
Polymerization/Depolymerization-Polymerization can generate mechanical force through the "Brownian ratchet" mechanism (32). Thermal fluctuations allow a polymer to make an energetically unfavorable step. The step is then "locked in place" by the next monomer added to the polymer, preventing relaxation to the previous state. If we apply the Brownian ratchet mechanism to strand exchange, we could imagine that, during four-stranded exchange, RecA covers dsDNA from the 5Ј end up to the Holliday junction. The junction now moves for-ward one step in the 3Ј direction by a thermal fluctuation, and the next RecA monomer snaps in place, preventing the return step. The Holliday junction would now move in a unidirectional way.
However, during in vitro experiments, the strand exchange reaction occurs under conditions where one of the substrates is known to be completely coated with RecA prior to the initiation of the reaction, while the RecA remains bound to the end product when the reaction is completed (17,33). This indicates that force transduction must be able to proceed without net polymerization or depolymerization.
Treadmilling-Treadmilling is a variant of the previous model in which there is no net polymerization or depolymerization (8). The treadmilling process occurs when monomers are removed from one end of a polymer segment and are added to the other end at the same rate. Treadmilling has been observed, for instance, in actin fibers.
The net effect would be that a RecA segment of roughly constant length travels along the DNA in the 5Ј to 3Ј direction. The Holliday junction could be imagined to be either at the front of the RecA-coated segment and propelled forward by it or behind the segment and dragged along with it (8). The segment could exert force on a junction by the Brownian ratchet mechanism in the same manner as before.
The fact that RecA monomers preferentially associate on one end of the nucleoprotein filament and preferentially dissociate from the other would be in support of treadmilling. The direction of motion of the junction is indeed consistent with that implied by the direction of RecA polymerization (8,25).
However, evidence of treadmilling in RecA has been sought for, but with no success (8). In particular, electron microscopy studies on nucleoprotein filaments do not exhibit the segmented structure expected from the treadmilling scenario (34). Additionally, changes in reaction conditions which are known to greatly affect the rate of RecA monomer association/dissociation have little or no effect on the rate of strand exchange, contrary to what would be expected if treadmilling were the propulsion mechanism for strand exchange (8).
Depolymerization/Repolymerization-The idea here is that RecA monomers sequentially dissociate from the invading strand in front of the exchange point, and re-associate behind the junction on the displaced strand as the exchange point moves forward by one monomer (35). The incoming monomer locks the Holliday junction in the new position, and the process begins again. Force transduction could proceed by the Brownian ratchet mechanism.
This model has several advantages. As in treadmilling, the total number of RecA monomers bound to the polymer remains constant. It also is consistent with the easier nucleation of RecA onto ssDNA, and it does not require monomer exchange with the bulk. However, the model is in disagreement with experiments on three-stranded reactions, in which the displaced strand is observed to be covered with single-stranded binding protein after the reaction is completed (instead of RecA), while the dsDNA product remains coated with RecA (17,33). The model also couples the strand exchange to depolymerization of RecA monomers from ssDNA. Although directional disassembly of RecA from ssDNA has recently been observed (36,37), it occurs at a rate of 105-120 bases/min. The observed strand exchange rate of about 380 Ϯ 20 bp/min (10) is more than 3 times this, and the depolymerization is therefore insufficient to explain it. Finally, the rate of strand exchange is insensitive to changes in reaction conditions which greatly affect the rates of polymerization and depolymerization (8).
Defect Removal-It has been proposed that ATP hydrolysis is only necessary to allow RecA monomers to dissociate and re-associate sufficiently in order to remove discontinuities in the RecA coating of the DNA (38). Strand exchange would derive the necessary energy for the reaction from the binding energy of the DNA within the nucleoprotein filament, i.e. by the lowering of the Gibbs free energy of the strands with no need for an additional energy supply.
Although possible for the three-stranded reaction, this model obviously cannot work for the case of the four-stranded reaction, where the reaction products are indistinguishable from the substrates, and where no net gain in free energy to drive the reaction is obtained from DNA binding. Having to invoke completely different force transduction mechanisms for threeand four-strand exchange reactions is not appealing.
Facilitated Rotation-Having ruled out force transduction by polymerization or depolymerization, we will now look for a force transduction mechanism that is more analogous to that of the motor proteins like myosin and dynein. The nucleoprotein filament is assumed static, insofar as monomer addition and removal are concerned, but the RecA monomer is able to generate force directly on DNA during ATP hydrolysis. The two DNA substrates are assumed to lie next to and parallel to each other during strand exchange. The DNA strand that is not covered by RecA binds to the exterior of the RecA helix. The RecA is assumed to act as a rotary motor that causes the two strands to be mechanically rotated around their axes, in an opposite sense, with the axes kept in place. Alternately, the two strands are rotated around each other (see Fig. 1). Strand transfer is then automatically accomplished by the rotary motion. Indeed, RecA has structural similarity with a well known linear rotary motor protein, F 1 ATPase (39).
This "facilitated rotation" model (20,25) avoids all the difficulties, mentioned above, that ruled out polymerization. Moreover, it is consistent with an elegant experiment by MacFarland et al. (40) discussed later, which shows that RecA is able to produce a rotary torque on DNA and transmit it along naked dsDNA over a considerable distance from the point where it is applied. However, the model also makes the following predictions. 1) The two substrates must lie parallel to each during strand exchange. 2) There must be DNA binding sites on the exterior of the RecA.
Little is known about the actual conformation of the Holliday junction when RecA coats one of the substrates. There is no incontrovertible experimental refutation of prediction (1), but there is much evidence which makes it seem very unlikely.
Work on Holliday junctions in which the DNA substrates are bare (41,42) shows that, in the presence of Mg 2ϩ (which is necessary for RecA-mediated strand exchange; Ref. 5), the Holliday junction adopts the configuration shown in Fig. 2 (43) with the small angle being about 63° (44). Although the undeformed or "helical" strands in Fig. 2 cross at an angle of 63°, the conformation is called "anti-parallel" because viewed from the side, the helical strands are parallel to each other, but with their 5Ј to 3Ј orientation reversed. A "parallel" conformation with the 5Ј to 3Ј orientation the same in both helical strands can also be seen in some circumstances (45). Rotating the near pair of diagonal arms in Fig. 2 counterclockwise by 60°produces the parallel arrangement.
The anti-parallel conformation shown in Fig. 2 is not in keeping with conventional models of recombination intermediates, because it implies that homologous regions of the substrates are separated from each other by significant distances (45). It would also make explaining any regions of triplex structure (22,27,46) formed during a strand exchange reaction very problematic. Finally, it would require the energetically expensive translation of the two substrates, roughly along their axes, with a relative velocity little less than double that with which the strand exchange occurs. This is in addition to the required rotation of the two substrates.
This makes it seem virtually certain that the anti-parallel conformation adopted by naked DNA Holliday junctions in the presence of Mg 2ϩ is not the conformation adopted during RecA mediated strand exchange. The parallel conformation, however, does not have the problems listed above. In addition, it has been shown that this conformation can be induced by effects at the crossover point (45), which implies that RecA may force the Holliday junction into this or a similar conformation. Since the angle which the groove on the RecA helix makes with the helix axis is about 50°, it is possible that the bare dsDNA molecule is somewhat constrained to be aligned with this groove.
It is difficult to judge how far the results from work on bare DNA can be applied to the RecA case, but so far as it applies, it implies that the substrates cross at a large angle. Certainly, work on bare DNA provides no evidence to indicate that the substrates in the recombination process are parallel in a geo-metrical sense or in contact with each other for an extended length.
Prediction 1 is also in conflict with some experiments. Analysis by chemical and nuclease probes indicate that there are probably no unpaired bases inside a Holliday junction (47). Although it might be argued that the RecA shields unpaired bases from nuclease probes, the only certain way to avoid this is by having the two double helices in the stacked helix conformation, either as shown in Fig. 2, or in the parallel conformation (45).
In summation, while we cannot rule out prediction 1, there is a large body of related information which makes us feel such a geometry is very unlikely. We will therefore assume that a correct model must be able to accomplish strand exchange when contact between the substrates is restricted to the point of exchange.
Regarding prediction 2, no DNA binding sites have so far been identified on the surface of RecA. This does not mean that such sites are not present, but the lack of any report of them in the literature is surprising. Additionally, electron micrographs generally do not show the substrates parallel to each other near the Holliday junction (24,27,38). It is reasonable to argue that electron micrographs are known to induce distortions, but it also seems reasonable to expect DNA bound to the exterior of the filament to be somewhat held in place by such binding if it exists.
Although these objections do not disqualify the facilitated rotation model from consideration, they do argue strongly against it. Nevertheless, despite these objections, the experiment mentioned above on transmission by DNA of RecA generated torques indicates that the facilitated rotation model addresses an important feature of force generation by RecA.
In summary, we will draw the following conclusions. 1) ATP drives the reaction. Limited three-stranded exchange reaction occurs spontaneously without ATP hydrolysis, but ATP hydrolysis is necessary for exchange of long strands. Four-stranded exchange strictly requires ATP hydrolysis in order to proceed. 2) ATP hydrolysis is cooperative. Hydrolysis of a RecA monomer-bound ATP triggers hydrolysis of the adjacent RecA monomer-bound ATP on the 3Ј side producing hydrolysis waves traveling 5Ј 3 3Ј along the filament. 3) Polymerization and depolymerization play no role during homologous strand exchange. 4) It is likely that torque is generated only at the strand exchange point. We assume that this is the case. This conclusion is based on our discussion above of the likely arrangement of the Holliday junction, and the lack of any evidence for DNA binding sites on the exterior of the RecA filament.

Hydrolysis Wave Model
Introduction-We have seen that a number of natural explanations of RecA force generation contradict one or more of the experimental observations. In this section, we will develop an alternative phenomenological model for RecA force generation, the hydrolysis wave model, based on the conclusions listed at the end of the preceding section. In addition, the hydrolysis wave model will work regardless of what the precise geometry is at the Holliday junction, and will generate torque only at the strand exchange point.
As in the facilitated rotation model, we assume that the RecA polymer always remains bound to DNA during strand exchange, and that polymerization and depolymerization play no role. Our model is based on the fact, discussed previously, that the RecA/DNA filaments exist in extended and collapsed forms (3), depending on whether the cofactor used is an NTP or an NDP. It follows that the RecA monomer must be able to adopt two quite distinct equilibrium conformations. One conforma- tion is adopted when the monomer is bound to ATP, the other when it is bound to ADP. The fact that enzymes can adopt multiple configurations, as proposed by Changeux, has been well established for many enzymes.
Suppose a RecA monomer is part of an extended filament and that the cofactor has just been hydrolyzed with no depolymerization taking place. Since it is known that it is not possible to directly interconvert between the extended and collapsed forms of the filament without depolymerization (4), the RecA monomer cannot adopt its equilibrium configuration; hence, it must be under some form of elastic stress: the "tense" state.
We know that hydrolysis is cooperative, with waves traveling from 5Ј to 3Ј (13). The front of this hydrolysis wave is expected to move with a velocity, , comparable to the depolymerization rate. Behind the wave front, there will be a region in which the RecA is in the tense configuration and either bound to an ADP cofactor or with no cofactor. It is known that RecA monomers spend only a small fraction of their hydrolytic cycle bound to ADP (1), indicating that the ADP region is likely to be rather short. The replacement of the ADP cofactor by an ATP cofactor from the bulk will allow the RecA monomer to revert to the starting configuration. Since strand exchange and torque generation depend on ATP hydrolysis (10), and assuming that contact is restricted to the strand exchange points, we conclude that the hydrolysis waves directly exert a force on the strand exchange point.
So far, we have not made assumptions that are not supported in some form by previous experiments. Our primary assumption is that the point of entry of the naked dsDNA into the nucleoprotein filament during three-stranded and fourstranded strand exchange, the exchange point, interacts attractively with the extended tense region behind the hydrolysis wave front. In other words, we assume that the free energy cost of producing an exchange point is less if the exchange point is inside the stressed region. In practice, the calculations presented below show that a repulsive interaction can have the same net effect as an attractive potential.
The result of such an attractive interaction between hydrolyzed sectors of the filament and the exchange point is that, for certain reasonable values of the physical parameters, the exchange point tries to bind to the hydrolyzed sector and is consequently dragged along by the traveling hydrolysis wave. For the case of a repulsive potential, the exchange point is pushed along in front of the hydrolysis wave. In both cases, the effect is to force the naked dsDNA into the nucleoprotein filament.
Three-stranded Reaction-The strand exchange process proceeds from here on the same as assumed by earlier workers. For the case of the three-stranded reaction, the dsDNA is brought into the filament and there is a region of the filament in which there are three strands of DNA inside the RecA helix. The newly introduced strands occupy sites II (for the strand complementary to the strand at site I) and III (for the strand identical to the strand at site I).
Recall that the binding of DNA to the different sites occurs with per monomer binding energies E I , E II , and E III with E III Ͼ E II Ͼ E I , so that the binding to site III is the weakest.
When the strand at site II switches its base pairing from the strand at site III to the strand at site I, the strand at site III is expelled from the filament interior (41,42,47). This means that it must be energetically unfavorable for the strand to remain in site III. There are two things which might contribute to this.
First, we speculated earlier that site III may be a repulsive anti-binding site with E III Ͼ 0. Second, DNA is negatively charged, and there will be electrostatic repulsion between all three strands. Both of these things would tend to expel the strand at site III from the filament interior. In both cases, the strand which occupies site III would originally have had to be dragged along into the filament by its base pair bonds to the strand which occupies site II, and the net process would need to be favorable, meaning that we require E III ϩ E II Ͻ 0.
It is interesting to consider the effect of polyvalent salts, such as Mg 2ϩ , on the strand exchange reaction. It is well known that polyvalent counterions facilitate DNA bundling (48). It appears reasonable to assume that polyvalent ions will stabilize triplex DNA/RecA complexes. Indeed, it is also well known that strand exchange reactions require Mg 2ϩ salt (38). It has also been established that Mg 2ϩ ions affect the angle of Holliday junctions on bare DNA, allowing for smaller angles (43). High salt concentration eases the assembly of the three-stranded structure and makes the eventual expulsion of the strand at site III more difficult.
Four-stranded Reaction-In the four-stranded reaction, the DNA initially in the filament is double-stranded and bound to sites I and II. The incoming DNA molecule intersects the filament, and the strand initially at site II is exchanged with the identical strand from the external DNA molecule. Our model can accommodate any angle of intersection between the incoming DNA molecule and the filament, but an angle of about 60°s eems likely, as this will allow strand exchange with no net breaking of Watson-Crick base pairs (47) (see Fig. 2). The incoming strand now occupies site II, from which it has just displaced the former strand. The point of strand exchange, the Holliday junction, interacts with a tense segment in a similar manner as the simple exchange point in the three-stranded reaction. The point of strand exchange is again propelled forward by the hydrolysis waves, and the strand exchange proceeds unidirectionally.
Like the facilitated rotation model, the hydrolysis wave model requires neither polymerization nor depolymerization. Both are essentially rotary motors and allow for the transmission of torque. The key mechanical difference with facilitated rotation is that, in the present case, the torque is generated at the exchange point and requires no extended region of substrate-substrate contact. There is thus no need in our model for the two substrates to run parallel for any length, nor for the presence of DNA binding sites on the filament exterior. Indeed, our model of the four-stranded exchange reaction predicts that a 60°intersection of the two substrates would be mechanically very favorable.

Quantitative Analysis
Rotary Force Generation-In this section, we will treat the physical aspects of force generation by the hydrolysis wave model in more detail. Our aim is first to show that the proposed mechanism really does act as a rotary motor and to compute its force-velocity relation. We will see that the model predicts that RecA differs substantially from conventional motor proteins. The second aim is to make quantitative predictions that should allow the model to be tested experimentally. Because some mathematical analysis is required here, it is suggested that readers not interested in the mathematical details skip to the concluding paragraphs of the section. Fig. 2 shows the geometry of a four-strand exchange reaction. A naked dsDNA strand makes a fixed angle with the RecA/dsDNA filament. If the angle is close to 60°, as in the anti-parallel arrangement shown in Fig. 2 or in the similar parallel arrangement, it will permit strand exchange without broken hydrogen bonds (47), as discussed earlier. Although the hydrolysis wave model will work with any angle of intersection, we will assume that the parallel conformation with an angle of approximately 60°is adopted. The RecA helical repeat length will be denoted by P. Strand exchange requires the naked dsDNA to slide along the groove of the RecA helix.
Geometrically, the strand exchange can be achieved in two different ways. 1) If the filaments are geometrically parallel and in the parallel conformation, the axes of the filament and the naked dsDNA strand can rotate around each other in space, maintaining a fixed relative angle; or 2) the direction of the strands remains fixed in space, but both strands rotate around their respective axes (like speedometer cables).
If the ends of the strands are constrained in space then process 1 is not possible. Likewise, process 1 will not work if the substrates are in the anti-parallel conformation. Process 2 requires that strands should be free to rotate to avoid build-up of torsional stress that eventually will stop strand exchange. We will focus on case 2, which is known to be the actual motion (23).
The dynamical parameters of the model are shown in Fig. 3. The front of an incoming hydrolysis wave is indicated by a time-dependent function x(t). Behind x(t), there are one or more tense RecA monomers that have just hydrolyzed their cofactor but have not yet exchanged their ADP cofactor with bulk ATP. We will assume that the bulk concentration of ATP is sufficiently high that this tense region is short (a few monomers).
Position y(t) denotes the strand exchange point along the RecA filament. As the RecA filament rotates, the exchange point moves forward. If ⍀ is the rotary angular velocity of the filament and P the pitch of the RecA helix, then the velocity Ѩ t y of the exchange point must equal P⍀/2. The distance between wave-front and exchange point is The key assumption of the model is that the free energy cost of the strand exchange point is significantly reduced if the exchange point occupies the stressed region behind the wave front. The free energy cost, U, of introducing an exchange point thus must depend on s(t) and we will denote it by U(s). If s(t) is large compared with the typical size S of the tense region (of the order of a few RecA monomers), then U(s)must be a constant (denoted by U 0 ) independent of s(t). For distances s small compared with S, the exchange point free energy is reduced below U 0 by an amount ⌬U. It is not possible to compute ⌬U without further detailed molecular modeling. Removing a RecA monomer from the filament right at the exchange point, however, would allow the two DNA strands to adopt an optimal configuration, so we may assume that ⌬U must be less than the free energy cost of removing a RecA monomer from the filament. Since ATP hydrolysis is known to produce spontaneous depolymerization, it is reasonable to conclude that ⌬U must be less than the ATP hydrolysis energy (about 12k B T).
The force generation mechanism relies on Newton's Third Law, according to which the exchange point and the wave front should exert equal but opposite forces on each other given by We now will construct the equations of motion obeyed by the exchange point and the wave front, starting with the latter case. The wave front is propagated forward along the filament through the cooperative nature of ATP hydrolysis. Let the activation barrier against hydrolysis of a single monomer inside a RecA/ATP filament be W. From the known cooperativity of the hydrolysis reaction, we know that this activation barrier must be significantly lowered once the neighboring RecA monomer on the 5Ј side has hydrolyzed its cofactor. Let WЈ be the reduced activation barrier for that case and let kЈ ϰ e Ϫ(WЈ/kBT) be the associated cooperative rate constant. The hydrolysis wave front has a "natural" velocity.
We may identify V ϱ with the depolymerization velocity.
The reaction rate will be altered if the wave front approaches a strand exchange point. We saw that the strand exchange free energy U(s) depends on the distance between wave front and exchange point, and this will alter the free energy gain of hydrolysis.
Suppose the wave front moves forward by one RecA monomer. The change in strand exchange free energy U(s) following a change of the separation s by one monomer length is ␦W ϭ a(␦U/␦s). This change in the free energy between initial and final states of the hydrolysis chemical reaction may or may not be reflected in the activation barrier.
There are two extreme cases. If the activation barrier is not affected by the change in the final state energy, then the reaction rate is just kЈ. If the change in final state free energy applies as well to the kinetic barrier, then we must replace WЈ in kЈ ϰ e Ϫ(WЈ/kBT) by WЈ ϩ ␦W. In general, the velocity of the wave front can be expressed as The length a 0 varies between 0 and a, the length of one RecA monomer, depending on the extent that this change in reaction free energy on altering s is reflected in a change of the activation barrier WЈ.
Next, we must construct the equation of motion for the exchange point. We saw already that the exchange point is subjected to a force F x (s) coming from the wave front. The exchange point will move, and the RecA filament will rotate around its axis, under this force. The resulting velocity of the exchange point is controlled by the rate of free energy dissipation. There are two contributions to dissipative losses: chemical dissipation due to base pair unstacking and breakage and reconstruction of pairs of hydrogen bonds right at the exchange point, and distributed hydrodynamic viscous losses due to rotary motion of the two strands.
It follows from the general principles of non-equilibrium thermodynamics that, at sufficiently low velocities of the exchange point, the free energy dissipation rate must be proportional to the square of the velocity of the exchange point and that the resulting drag force F() must be proportional to . We will confine ourselves to this regime of linearized non-equilibrium thermodynamics.
The proportionality constant in the relation F() ϭ / is known as the mobility. The energy dissipation rate is 2 /. If viscous losses dominate, then the hydrodynamic mobility can be computed explicitly (49) with the viscosity of water (0.01 in cgs units), L the length of the strand, P the repeat length, and R the radius of the filament (or DNA strand), treated as a cylinder. We have used the fact that is related to the rotation rate by P⍀/2. The mobility associated with base pair unstacking and the fracture and reconstruction of hydrogen bonds cannot be computed explicitly without microscopic modeling. Applying the phenomenological Kramer's rate theory (see Ref. 50, and references therein) of thermally activated chemical reactions, however, gives the following expression Here, a bp is the base pair spacing, and is the attempt frequency. The attempt frequency is the number of attempts per second made by a molecule to escape out of a meta-stable state. It is related to the reaction rate K by K ϭ Ϫ⌬F/kBT with ⌬F the free energy cost of the transition state as compared with the metastable state. It is often difficult to obtain accurate estimates of , but if we assume that the relevant transition state moving the Holliday junction has an extra broken base pair, then would be a typical vibrational frequency of the DNA bases (of the order of 10 9 /s). The transition state energy W bond is then the duplex formation energy per base pair, about 13k B T (51). Note that the mobility is quite sensitively dependent on this activation barrier, which is not known, so it is difficult to estimate chem . Using these values for W bond and , we would obtain a mobility in the range of 10 3 m/N s.
The total mobility, including both effects, is the inverse of the sum of the inverses of the two mobilities. For reasonable parameter values (strand lengths of the order of microns), "chemical" dissipation rates are huge compared with hydrodynamic dissipation rates and we can safely neglect the latter. To obtain typical strand exchange velocities in the measured range of 6 -7 bp/s (10), the relation F() ϭ / requires that the force exerted on the exchange point by RecA hydrolysis be in the range of 0.1 pN, if we assume a mobility of 10 3 m/Ns.
We have seen that a nearby wave front exerts a force on an exchange point, but a retarding torsional stress, generated by the rotary motion, could very well be present, as well as other forms of external forces. In the future, when micromechanics experiments on Holliday junctions become possible, externally applied forces will be adjustable experimentally, allowing measurement of the "force-velocity" curve. We will denote the total external retarding force F Ext . The equation of motion of the exchange point is now We can find a single equation of motion for the relative spacing s by subtracting Equation 8 from Equation 5, giving where we defined here an "effective" potential energy of interaction.
kBT ѨsU͑s͒ͮ ds (Eq. 10) Note that U eff (s) does not have the dimension of energy. The motion of an exchange point under the impact of a hydrolysis wave front is now fully determined by the two preceding equations.
Assume that a hydrolysis wave approaches a strand exchange point from the 5Ј side. Mathematical analysis shows that two scenarios are possible: either the exchange point is "captured" by the hydrolysis wave front, and the two points move with a common velocity of the order of that of a hydrolysis wave front; or the hydrolysis wave front overtakes the exchange point, giving it a "kick" in the process, and continues on. In the first case, the motion of the exchange point will be smooth. In the second case, the exchange point will move in a "jerky" manner under the repeated impact of hydrolysis waves moving from the 5Ј to the 3Ј end.
Trapping-The first case requires that the effective potential energy of interaction U eff (s) has a minimum, denoted by s*. The reason is that for the wave front to capture the exchange point, we must demand that Ѩ t s ϭ 0, and since Ѩ t s ϭ ϪѨ s U eff (s), the first derivative of the potential must vanish. This is the condition that the effective potential has an extremum. Only a minimum of the effective free energy would correspond to a physically stable configuration. A necessary (although not sufficient) condition for the existence of a minimum is thus that the equation Ѩ s U eff (s) ϭ 0 has a solution. We can write this condition as the requirement that F Ext ϭ F ͑s*͒ Ϫ Ṽ e ϪF ͑s*͒ (Eq. 11) must have a solution. Physically this is just Newton's First Law: the external retarding force on the strand exchange point must equal the driving force exerted by the wave front on the exchange point minus the drag force on the exchange point (which is also the dimensionless velocity of the exchange point).
We have introduced here three dimensionless quantities. The dimensionless external force is given by F Ext ϭ a 0 F Ext /k B T, the dimensionless hydrolysis velocity is defined by Ṽ ϭ a 0 V ϱ / k B T, and the dimensionless force exerted by the wave front on the exchange point is given by F (s*) ϭ a 0 (dU/ds)͉ s* /k B T. For conventional motor protein systems, these dimensionless parameters are usually of the order of 1 (32).
For a potential profile consisting of a simple well with a depth ⌬U and a width S, there can be two solutions, one solution, or no solution for equation (11). In the first case, the first solution (i.e. with the smallest value of s*) is a minimum of the effective potential while the second solution is a maximum.
Assuming that a solution exists, we can use Equation 11 to compute the relation between the dimensionless velocity of the exchange point Ṽ exch ϭ Ṽ e ϪF ͑s*͒ (Eq. 12) and the external force, producing a force-velocity curve In Fig. 4, we have plotted the dimensionless force-velocity curve for a dimensionless hydrolysis velocity of 4. Maximum Retarding Force and Disruption-It would seem that we could apply larger and larger external forces with the exchange point merely reducing its velocity. That is of course not possible. The maximum value of the dimensionless force applied by the wave front is of the order of where S is the characteristic size of the potential trap (of the order of a few monomer lengths), and ⌬U is the depth of the trap. If we use the earlier estimates, then this dimensionless maximum force is of the order of 1-10. It would seem reasonable that the maximum external retarding force beyond which trapping becomes impossible should equal F hyd (max). According to Equation 11, however, the actual maximum retarding force is less than F hyd (max) because of the drag force Mathematically, this is precisely the point where the two solutions of Equation 11 fuse and then disappear. Note that increasing the hydrolysis rate reduces the threshold force required to break the connection between the wave-front and the strand exchange point. The lower this threshold force, the easier force transduction by hydrolysis is disrupted by external forces. The velocity of the exchange point right at the point of disruption is Solving for the exchange point velocity, we find the following result At low hydrolysis rates, the maximum velocity is thus equal to that of the hydrolysis wave front. At higher hydrolysis rates, the exchange point moves slower than the natural velocity of the wave front. In fact, the maximum velocity of the exchange point in dimensionless units cannot be large compared with 1, since ln(Ṽ ) is of the order of 1 even if Ṽ Ͼ Ͼ 1. In standard units, we can conclude that V exch Ϸ Ͻ k B T/a 0 . This is interesting because it means that the maximum velocity of the exchange point is directly determined by the mobility of the exchange point. Note also that the force F Ext exerted on the exchange point is of the order of k B T/a 0 in this case since V exch Ϸ Ͻ F Ext /. Interestingly this is just the characteristic force of a Brownian ratchet (30), even though the hydrolysis wave model is not a Brownian ratchet.
Stochastic Motion-If the external force exceeds F Ext (max), then the exchange point can no longer be trapped. If there is no wave front in the neighborhood, then the exchange point drifts with a velocity F Ext / along the negative x direction. When a wave front does pass by, it tries to drag the exchange point along in the positive x direction, but eventually the exchange point is left behind. The cycle is repeated as the next wave front passes by from the 5Ј to the 3Ј direction. The net result is a stochastic "back-and-forth" motion.
We will consider the case that the external force is just a bit larger than the maximum force, F Ext (max), and assume that wave fronts pass at a certain rate ⌫ along the filament. As a wave front approaches an exchange point, the exchange point will start to move. The center of mass motion will be fastest at the point where the force exerted by the wave front on the exchange point is at a maximum but the relative motion is slowest at that point. We expand the effective potential at the point of maximum force s max as Direct integration shows that the exchange point remains inside the tensed region behind the front for a trapping time T of the order of During the trapping time, the exchange point moves with a velocity of the order of Ṽ exch (fracture) ϭ Ṽ e ϪFhyd(max) . The exchange point is dragged along the positive x direction a distance T(F Ext ) Ṽ exch (fracture). Following release, the exchange point must wait a time ⌫ Ϫ1 for the next wave front to arrive. In this "free" time, the exchange point moves a distance ϪF Ext /⌫ along the negative x direction. The average velocity is and using the result for the trapping time gives a force-velocity curve, The complete force-velocity curve has a sharp cusp at the fracture point. For large enough external force, the average velocity reverses sign. The critical external force where the velocity is equal to zero is known as the "stall-force." We can now draw the full force-velocity curve (Fig. 5).
The exchange point velocity drops very rapidly when the external force exceeds the fracture force. For lower rates ⌫ of hydrolysis wave production, we can identify the stall force and the fracture force, as follows from Equation 24. As the rate increases, the stall force will start to exceed the fracture force. The well defined "kink" in the force-velocity curve is a feature that is absent from conventional motor protein models and should act as a clear indicator for the validity of the description proposed in this paper.
We expect that the exchange reaction will take place with a very constant and predictable velocity determined principally by the depolymerization velocity depol , but slowed in a predictable and calculable manner by the drag introduced as a result of having to push the exchange point forward, with the added necessity of rotating and translating the two substrates in order to do so.
Unwinding of an Unpaired Distal Segment of One Substrate-MacFarland et al. (40) have reported an experiment in which circular dsDNA with a single-stranded gap was coated with RecA and underwent a four-stranded exchange reaction with a linear dsDNA molecule not coated with RecA. They conducted several experiments in which the linear DNA substrate did not extend to the end of the double-stranded region of the circular DNA, and reported that separate hybrid DNA molecules were formed.
They interpreted this as indicating that the distal region of the circular dsDNA was unwound by the reaction as far as the single-stranded gap, allowing the products to separate. The presence of a nick in the gapped strand of the circular DNA substrate terminated the unwinding at that point, and the two products separated. The products in either case were a hybrid circular dsDNA molecule with a single-stranded gap and a linear duplex hybrid DNA molecule with a single-stranded tail at the distal end.
Although MacFarland et al. interpreted these results as support for the facilitated rotation model and a refutation of models requiring monomer rearrangement, it is clear that these results are what would be expected from the hydrolysis wave model as well. The process of expelling the displaced strand would continue, in the presence of RecA hydrolysis, even after there was no corresponding strand to replace it because the point at which a DNA strand leaves the nucleoprotein filament would be bound to the ADP region in the same manner as the point at which a DNA strand enters the filament. This continued, forced expulsion of the DNA would naturally unwind the remaining DNA.
A slight variation on this experiment, however, would distinguish between the hydrolysis wave model and the facilitated rotation model. This unwinding should occur for the facilitated rotation model, regardless of whether the extra distal extension is on the RecA-coated molecule or on the bare DNA molecule. In the hydrolysis wave model, however, the distal extension will only be unwound if it is on the RecA-coated molecule. If it is on the other molecule, it is not within a nucleoprotein filament, and since unwinding is accomplished by expulsion from the filament, it cannot then occur.
Consider then an experiment similar to that of MacFarland, where now the linear dsDNA molecule has the single-stranded primer at the proximal end, and the circular DNA substrate merely has a nick in one strand at the point where the unpaired FIG. 5. The full force-velocity curve, including regions beyond the fracture force, when externally applied force is too great for trapping to occur. Note the sharp kink in the force-velocity curve when the fracture force is reached, and the rapid drop in velocity as the force is further increased. distal end extension meets the point at which the strand exchange begins. In the facilitated rotation model, the linear molecule should continue to be counter rotated around the RecA-coated molecule after the homologous pairing is completed, and the distal segment should be unwound. If the hydrolysis wave model is correct, however, there is no expulsion of the displaced strand since there is no RecA coating the unpaired distal segment, so that the products should not separate and will remain as a joint molecule. CONCLUSION Summary-We propose that the mechano-chemistry of the RecA force transduction mechanism is unique: it relies neither on polymerization/depolymerization (as observed for actin and microtubule strands) nor on direct force transduction by the motion of flexible sections of the protein as for conventional motor proteins (like kinesin, dynein, and myosin). Instead, force is generated by localized, solitary hydrolysis waves propagating along the RecA helix, which impact the DNA exchange site. The proposed model is consistent with the experimental information on RecA force transduction available to date, including both the most likely structure of the exchange site as well as the fact that experimental conditions that alter polymerization behavior do not affect the rate of strand exchange.
Experimental Tests-The central features of the proposed model are open to experimental testing. The computed forcevelocity curve shows a sharp kink at the point where the retarding force is strong enough to cause the strand exchange point to break away from the hydrolysis wave. Force-velocity curves of RNA polymerase, myosin, and other proteins have been determined experimentally by attaching the protein to a bead that could be trapped by a laser. The analog of these experiments would be to apply a rotary torque on the DNA strand that could either advance or retard the motion of the exchange site. Rotary torques have been applied to DNA by the magnetic bead method and could be employed for a study of strand exchange force transduction (52).
A second test would be an extension of the experiment of MacFarland et al. (40). The substrates would be a linear dsDNA molecule with a single-stranded primer at the proximal end, and a circular DNA substrate with a nick in one strand. This should result in the linear DNA molecule being the RecAcoated substrate, unlike the MacFarland experiment in which it was the circular DNA molecule which was coated with RecA. The circular DNA molecule should have a distal end which extends further than the end of the linear DNA substrate, and the nick should be between this unpaired distal end extension and the point at which the strand exchange begins.
The facilitated rotation and hydrolysis wave models make different predictions for what happens next. The facilitated rotation model predicts that the substrates should continue to be counter-rotated after the homologous pairing is completed, so the distal segment should be unwound to produce two separate molecules. The hydrolysis wave model predicts that the products should not separate but remain as a joint molecule.
Open Questions-We have examined in this paper only the mechanism of mechano-chemical force transduction for homologous DNA. We have not addressed one of the central functions of the force transduction mechanism: heterology bypass. Threestrand exchange proceeds spontaneously (in the absence of heterologies) since the exchange process lowers the free energy. Four-strand exchange does not lead to any net lowering of the free energy and requires ATP hydrolysis. DNA strand exchange between two organisms is only biologically relevant if there are differences in the base pair sequences of the two strands.
The length of heterology needed to interfere with successful strand exchange has been studied (53)(54)(55)(56). There is some contradiction within the literature, but heterology bypass has been reported to have a dependence on both the length and the location of the heterology. It would be of great interest to develop a theoretical model which both describes how the exchange process solves topological problems encountered during heterologous exchange and is consistent with the observed effects which length and location of the heterology have on strand exchange. It is also interesting to speculate how the proposed mechanism for strand exchange could have evolved. Force transduction is not crucial for three-strand exchange (14,24). If the original function of RecA was to repair DNA through a threestrand exchange process, then the main function of ATP hydrolysis was to provide a means of removing the RecA helix from the repaired strand (9). Hydrolysis waves with no depolymerization would assist the exchange process. When the RecA helix was recruited to promote four-strand exchange, the secondary function of force transduction took center stage and provided a means of overcoming the inevitable heterologous inserts.