In our system, we use Optical Tweezers to apply and measure forces on individual dsDNA molecules. It consists of a near infrared laser beam (SDL 5422H1, 830 nm wavelength) focused through a 100X objective (Zeiss, 1.25 numerical aperture, oil immersion) together with a Quadrant Photodiode (QPD) that monitors the position of a trapped microbead (Figure 1). The power of the laser beam at the optical trap is about 30 mW. Single dsDNA molecules are attached at one end to the bottom cover glass and at the other end to a polystyrene microbead of 1.6 μm diameter (Polysciences). The experiment is performed in a flow cell that allows changing the biochemical composition of the sample. To avoid breaking the bead-dsDNA-cover glass constructs, we inject new solution at relatively low flow rates using a DC motor (Newport) to activate a syringe. Following the addition of RecA to the sample and its assembly on the dsDNA molecules, the dsDNA elongates by an extent proportional to the amount of bound protein. To monitor the change in the contour length of a particular dsDNA molecule, we extend the filament by applying an approximately constant force on the trapped bead (∼0.8 pN).

The force exerted by the dsDNA-RecA filament on the trapped bead, F, manifests as a shift in the bead position with respect to the center of the trap, Δx. Correspondingly, the value of Δx affects the distribution of the laser light scattered from the trapped bead. This variation in the distribution of the laser light transmitted through the sample is monitored by the QPD. Calibrating the QPD allows to deduce the value of Δx from the voltage difference between the appropriate quadrants. Although the system allows a sampling rate of 20 KHz, we average the data down to rates in the 1–10 Hz range to reduce the effect of Brownian motion. The force is adjusted whenever it deviates from the 0.8 pN mark by an appropriate shift of the sample. Since the optical trap is at a fixed position, we move the sample to change the end-to-end distance of the DNA molecule, leading to a corresponding change in the force. A Peltier element together with a feedback control unit (PID) is used to maintain a constant temperature in the sample at 37.0 ± 0.1°C. It is attached to a copper ring that is in thermal contact with the microscope objective.

The bead-dsDNA-glass construct is prepared using a low pH protocol for spontaneous binding (Allemand et al., 1997; Shivashankar et al., 1999). λ-DNA, 48.5 Kb, 16.5 μm (Promega) is incubated with the beads and PBS buffer at pH = 4 for 15 min to obtain the DNA-bead link. Next, the solution of DNA and beads is injected into the sample and after 2 h incubation we find that a certain fraction of the DNA molecules are tethered both to the glass bottom and to a microbead. Finally, we gently wash the sample to remove free beads and unbound DNA molecules. For a particular bead-dsDNA-glass construct, we verify that the bead is tethered by a single dsDNA molecule by measuring its force extension behavior and comparing it to the prediction of the Worm Like Chain (WLC) model (see Eq. 1) (Marko and Siggia, 1995). We then stretch the dsDNA and inject the protein together with the appropriate buffer solution into the sample cell. The final concentrations are as follows: RecA (9.33 μM), ATPγS (4.5 mM), MgCl₂ (6.25 mM), DDT (6.25 mM), TrisHCl (18.75 mM), and the pH is 7.9. The pH and the protein and buffer concentrations were chosen such that the nucleation is sufficiently slow allowing to observe the dynamics due to individual nucleation events.

Although the low pH protocol for obtaining the bead-dsDNA-glass construct is not widely used in single molecule experiments, it is particularly straightforward allowing us to perform a relatively large number of experiments. On one hand, its main drawback is that it is by far less specific than, for example, the standard biotin:streptavidin tethers, leading to a fraction of bead-dsDNA-glass constructs where the dsDNA is attached at some internal site rather than at its end to either the glass or the bead. However, for our experimental approach, such constructs are equally suitable as those where the dsDNA is only tethered at its ends. To include the non-specific constructs in our experimental data, we measure the contour length of the dsDNA between tethering points for each construct using the WLC model (see Eq. 1) (Marko and Siggia, 1995). Moreover, we test that the length of the dsDNA is not affected by the increase in pH to 7.9 in preparation for the RecA reaction. On the other hand, the non-specific constructs allow obtaining data for different contour lengths of the naked dsDNA, an additional parameter that affects the polymerizarion dynamics of RecA on dsDNA.

To monitor the length dynamics of the dsDNA-RecA complex during the polymerization process, we need to find the way it is related to the measured time dependence of the force exerted on the trapped bead. The equilibrium behavior of polymers under tension has been extensively studied (Smith et al., 1992; Perkins et al., 1995; Cluzel et al., 1996; Simmons et al., 1996; Wang et al., 1997). Using the WLC model, it was shown that the force, F, required to stretch the polymer to a certain end-to-end distance, z, is linear in the small z regime and rapidly grows as z approaches the contour length, L (Marko and Siggia, 1995). A good approximation to the exact F ( z ) is provided by the interpolation formula F ( z ) = k B T A [ z L + 1 4 ( 1 − z / L ) 2 − 1 4 ] (1) where A is the persistence length, T, the temperature and k B , the Boltzmann constant.

For naked dsDNA in solution of physiological ionic strength and pH ≈ 7 , it was found that the persistence length, A 0 , is about 50 nm. The effect of the RecA binding on the force-extension behavior is twofold: 1. the RecA-dsDNA complex is longer that the naked DNA, leading to an increase in the contour length, L ( t ) , as more RecA assembles on the dsDNA, 2. the RecA-dsDNA filament is significantly more rigid than the naked dsDNA molecule, A R ≫ A 0 , where A R is the persistence length of RecA-dsDNA. At intermediate stages of the RecA-dsDNA filament assembly, several domains on the dsDNA are decorated with protein while the rest is naked. In the large extension regime, F ≫ k B T / A , one expects that the force-extension behavior of a partly decorated filament only depends on the decorated length fraction, ϕ, rather than on the specific partition into domains. Accordingly, the relation between force and extension in partially assembled filaments is equivalent to that of a filament with a single RecA-dsDNA domain starting at one of the ends of the dsDNA and the same value of ϕ (Figure 2). Moreover, in the F ≫ k B T / A regime, we can neglect the first and last terms of Eq. 1 and characterize each of the decorated and the naked sections by their corresponding contour length, persistence length and end-to-end distance, L R , A R , z R and L 0 , A 0 , z 0 , respectively (Figure 2). Clearly, F = F R = F 0 , L ( 0 ) = ( 2 / 3 ) L R + L 0 and z = z R + z 0 , where F, F R and F 0 are the forces on the entire mixed filament, on the protein decorated domain and on the naked dsDNA part, respectively. Also, L ( t ) is the contour length and z, the end-to-end distance of the entire filament. Since in our experiment we measure F ( t ) , L ( 0 ) , Z ( t ) and A R , these relations correspond to four equations with four unknowns, L R , L 0 , z R and z 0 . Solving these equations provides an expression for the contour length of the protein decorated portion of the molecule L R = z − L ( 0 ) ( 1 − C 0 ) ( 1 / 3 ) + ( 2 / 3 ) C 0 − C R (2) where C 0 = k B T / 4 F A 0 and C R = k B T / 4 F A R . Using the appropriate values for the persistence lengths, A 0 = 50   nm and A R ≈ 1200   nm , together with the value of L ( 0 ) that is measured at the start of each experiment, we can use Eq. 2 to extract the contour length of the RecA-dsDNA domain from the measured values of F ( t ) and z ( t ) . Since the total contour length is simply related to L R , this allows obtaining the time dependence of the dsDNA decorated fraction, ϕ ( t ) . While for the persistence length of the naked dsDNA, A 0 , at the conditions of our experiment, we use the standard 50 nm value (Smith et al., 1992; Rocha et al., 2004), we have measured the value of A R for dsDNA molecules that were fully covered by RecA. Although we found some variation between the A R values obtained for different molecules, the 1,200 nm value represents a good approximation to the persistence length of the RecA-dsDNA complex whenever the RecA coverage is complete.

In the regime where the rate of protein nucleation, k n , is much larger than the rate of domain growth, k g , we may assume that the number of domains, N, is a continuous variable. This allows obtaining a model that links the dynamics of N ( t ) with that of the DNA coverage, ϕ ( t ) , via a set of two coupled differential equations (Avrami, 1939; Shivashankar et al., 1999) d N d t = k n ( 1 − ϕ ) − k g N 2 ( 1 − ϕ ) (3a) d ϕ d t = k g N (3b) While the rate constants of Eqs. 3a,b depend on the number of RecA binding sites on the naked DNA, L 0 / a , where a is the length of a RecA binding site (3 bp), these can be normalized to obtain the corresponding microscopic values, n and v , such that k n = n L 0 / a and k g = a v / L 0 . n is the nucleation rate density (per unit time per binding site) and v is the average growth velocity of individual domains. While the first term of Eq. 3a describes the creation of new nuclei at a rate proportional to the undecorated part of the dsDNA molecule, the second term depicts the reduction in the number of domains due to collisions between their fronts. The model assumes periodic boundary conditions on the dsDNA molecule (circular DNA). However, the model also assumes a large number of domains, N ≫ 1 , and since N has to be much larger than the total number of DNA base pairs, L 0 / a , we are in the limit of large L 0 / a where boundary effects are negligible. In other words, the model of Eqs. 3a,b is equally accurate for both circular and linear DNA.

Equations. 3a,b can be easily solved leading to a sigmoidal behavior for the dynamics of the protein coverage ϕ ( t ) = 1 − exp ( − 1 2 k n k g t 2 ) (4) Moreover, the model of Eqs. 3a,b can be generalized to include the case where the average rate of growth of the domains is asymmetric, such that, the growth rate in the 3′ to 5′ direction of the dsDNA is r times slower than that in the reverse direction. For this case ϕ ( t ) = 1 − exp ( − 1 2 k n k g ′ t 2 ( r + 1 ) ) (5) where k g ′ is the rate constant of the fast front such that the growth rate of the domain is k g = k g ′ ( r + 1 ) . Note that, unlike k g , k g ′ represents the growth rate of individual fronts rather than that of entire domains. Since the behavior of the protein coverage in the fast nucleation regime (Eqs. 4, 5) depends only on the product of the reaction rates, k n k g , and the front propagation asymmetry factor, r, one cannot obtain each of these parameters separately by comparing between the model and the corresponding experimental measurements. Instead, to determine each of these parameters separately, the assembly of the protein on dsDNA needs to be analyzed in a regime where nucleation is not much faster that the growth rate.

Another regime of the nucleation and growth process that can be described via an exactly solvable model is the limit of slow nucleation, k g / k n ≫ 1 . Here, we may assume that there is only one domain that grows to cover the entire dsDNA molecule. Since the location of the nucleation site for this domain can be anywhere along the dsDNA, the dynamics of the coverage, ϕ ( t ) , is different for each realization. However, in all realizations, the coverage dynamics is bilinear. While, the first slope corresponds to the time before the first front reaches the end of the molecule, the second slope represents the time interval until the other front reaches the other end of the dsDNA φ ˙ ( t ) = ( r + 1 ) k g ′ − k g ′ θ ( t k g ′ − x ) − k g ′ r θ ( r t k g ′ − ( 1 − x ) ) (6) where x · L is the position of the nucleation event and θ ( x ) is the step function.

Averaging Eq. 6 over all possible realizations, gives the ensemble averaged coverage dynamics, ϕ ¯ ( t ) , (Turner, 2000) ϕ ¯ ( t ) = { k g ′ ( 1 + r ) t − 1 2 k ′ g 2 t 2 ( 1 + r 2 ) t < 1 k g ′ k g ′ r t − 1 2 k ′ g 2 r 2 t 2 + 1 2 1 k g ′ < t < 1 r k g ′ 1 t > 1 r k g ′ (7) Unlike the behavior in the fast nucleation regime, here ϕ ¯ ( t ) depends separately on the values of k g ′ and r and does not depend on k n . This can be used to determine both k g ′ and r by fitting the behavior of Eq. 7 to the experimentally measured coverage dynamics in the low nucleation regime. Moreover, here asymmetric domain growth leads to a three step behavior for ϕ ¯ ( t ) instead of the two step behavior of the symmetric case.

We have performed a series of experiments under the same conditions as those leading to the results of Figures 3–5 (see Supplementary Section S1 for details). As one would expect, the L ( t ) kinetics that was measured is quite different from one experiment to another. However, in most experiments we observe the step-like structure reminiscent of that of Figure 4. Using the experiments where the number of nuclei is not too large, we repeated the analysis described in the previous section to obtain the full domain kinetics. Although in some of the experiments the domain kinetics was either incomplete or could be only partially analyzed, we also used such situations to extract the front velocities for some of the domains. For the latter, it is often difficult to distinguish between the velocities of the fast and slow fronts. Accordingly, we have used all available data to obtain the total velocities, v i = v i s + v i f , for 35 different domains. The distribution of these v i ′ s , P ( v i ) , is displayed in Figure 7. The corresponding average, v ¯ = 9 ± 1   RecA   sec − 1 , leads to a domain growth rate, k ¯ g = a v ¯ / L ¯ 0 = ( 7 ± 1 ) ⋅ 10 − 4 ⋅ sec − 1 .

The value of the average domain growth rate, k ¯ g , that was obtained from Figure 7 cannot be measured in the many nuclei regime of Eqs. 3–5, since the L ( t ) kinetics in this regime is fully determined by the k n k g product. This suggests that in the few nuclei regime, where our study was performed, we can also establish the corresponding value of the average rate of nucleation, k ¯ n . To this end, we use the relation between the extent of undecorated dsDNA and the probability of a new nucleation event Δ p ( t ′ < t < t ′ + Δ t ) = n L f r e e ( t ) Δ t (9) where Δ p ( t ′ < t < t ′ + Δ t ) is the probability of forming a new nucleus in the ( t ′ , t ′ + Δ t ) time interval, Δ t represents an infinitesimal time step and L f r e e ( t ) is the part of dsDNA not covered by protein at time t. Integrating Eq. 9 between one nucleation event at t i and the next at t i + 1 , the left side becomes equal to 1 and on the right side we obtain the integral of L f r e e ( t ) between t i and t i + 1 times n 1 = n ∫ t i t i + 1 L f r e e ( t ) d t (10) Moreover, the relation of Eq. 10 can be expanded to include the first N nucleation events, namely, N = n ∫ t 0 t N L f r e e ( t ) d t = n B N (11) where t 0 is the time when the protein was added to the sample, B _N denotes the time integral of L f r e e ( t ) from t 0 to the time when the Nth nucleus is formed, t N , and L f r e e ( t ) is directly related to the measured L ( t ) , L f r e e ( t ) = 3 L ( 0 ) − 2 L ( t ) .

Equation 11 allows us to estimate the average nucleation rate density, n ¯ . This is achieved by computing the values of B _N for each of the different experiments and obtaining the corresponding averages, B ¯ N . Then, according to Eq. 11, the slope of the plot of B ¯ N vs. N is n ¯ − 1 (see Figure 8). The best linear fit to the B ¯ N vs. N data shown in Figure 8 leads to n ¯ = ( 13 ± 4 ) 10 − 8   binding   site − 1 ⋅ sec − 1 , which, in turn, corresponds to an average nucleation rate, k ¯ n = ( 1.5 ± 0.5 ) ⋅ 10 − 3 ⋅ sec − 1 .

For three of our experiments, the domain kinetics inferred from our model allowed to obtain the velocity of each of the individual fronts. In these experiments, the L ( t ) kinetics were complete, had well separated nucleation times and their decomposition was unambiguous. Of these three experiments, one had two nuclei and its full analysis was presented in the previous section. The other two experiments had one and three nuclei, respectively, leading to a total of six nuclei for which we can estimate, r, the velocity ratio between the slow and the fast fronts, r ¯ = 0.25 ± 0.08 .

While in the few nuclei regime where we have performed our experiments we can separately measure the average values of the nucleation and growth rates, k ¯ n and k ¯ g , the theoretical description of the protein assembly process in this regime cannot be described by the analytical models presented in the Materials and Methods section, Eqs. 3–7. Instead, we need to use a nucleation and growth model that describes the formation of a few nuclei at random positions along the dsDNA and the dynamics whereby the emerging protein domains expand to cover the entire dsDNA molecule. We use numerical simulation to predict the average domain kinetics for our model and compare the outcome to the average experimental L ( t ) . This comparison also provides an alternative method to measure the values of the average nucleation and growth rates, k ¯ n and k ¯ g .

In the intermediate regime, where k n and k g are of the same order, we describe the nucleation and growth of the RecA on dsDNA using a Monte Carlo type model (Turner, 2000). It consists of two components: 1. nucleation of a protein monomer at a random position, x _i , along the (0, L) interval, and 2. growth of the ith domain starting at x _i and growing with rate k g ′ at one end and r k g ' , 0 ≤ r ≤ 1 , at the other. The time step for the simulation, δ t , was set to be 0.5 s, sufficiently small to ensure that the resulting kinetics does not depend on δ t . At each time step, a nucleation event will occur with probability k n ( 1 − ϕ ) δ t at a random location along the undecorated part of the dsDNA. In addition, each of the existing domains will grow by k g ′ δ t at its fast front and by k g ′ r δ t at its slow front. The side of the ith domain corresponding to the fast front is randomly chosen at t i , its nucleation time. The simulation is stopped when the protein covers the entire dsDNA molecule, ϕ = 1 , except for the case when r = 0 where full coverage occurs after very long time (Figure 9). To obtain the average kinetics of the protein coverage, ϕ ¯ ( t ) , we repeat the simulation M times and average ϕ ( t ) over all the runs ϕ ¯ ( t ) = 1 M ∑ l = 1 M ϕ l ( t ) (12) The fluctuations of ϕ l ( t ) between one run and another are quantified by the corresponding standard deviation, σ ( t ) , σ 2 ( t ) = 1 M ∑ l = 1 M ( ϕ l ( t ) − ϕ ¯ ( t ) ) 2 (13) To obtain the behavior of ϕ ¯ ( t ) in the different parameter regimes, we have used a relatively small ensemble, M = 10 3 . At this level of averaging, the corresponding error of ϕ ¯ ( t ) , σ ( t ) / M , is too small to be graphically resolved (Figure 9).

In Figure 9 we show the behavior of the average coverage, ϕ ¯ ( t ) , for both the slow and fast nucleation regimes and compare the numerical results to the corresponding theoretical predictions of Eqs 5, 7. In both these equations, time appears multiplied by the domain growth rate, k g = ( 1 + r ) k g ′ , that sets the time scale for the kinetics in the different regimes. Therefore, in order to compare the kinetics of the nucleation and growth process at different values of the parameters it is necessary to normalize the time by the corresponding time scale τ g = k g − 1 . Moreover, we found that the ϕ ¯ ( t ) kinetics for a particular growth asymmetry, r, only depends on the k g / k n ratio and not on k g and k n separately (not shown). To examine the effect of the growth asymmetry on the nucleation and growth process for each regime, we fixed the nucleation rate, k n , and varied the values of r and k g ′ , such that k g remains constant (Figure 9).

For the slow nucleation regime, the larger the growth asymmetry, smaller r, the slower will be the process of decorating the dsDNA on the side of the slow front of the first domain. Consequently, for small r, the ϕ ¯ ( t / τ g ) increases at a slower rate and the RecA proteins are less efficient in covering the dsDNA (Figure 9A). Moreover, while Eq. 7 only describes the growth kinetics for a single domain, at small r the undecorated fraction of the dsDNA at a particular time is larger, leading to a larger probability for a second nucleation event to occur. The contribution of the i > 1 domains to ϕ ¯ ( t / τ g ) leads to a growing discrepancy between the prediction of Eq. 7 and the results of the simulation as r decreases .

The lower efficiency of protein coverage for asymmetric growth is also found in the fast nucleation regime (Figure 9B) and can be ascribed to a mechanism similar to the one presented in the previous paragraph. When the number of domains is not too large, asymmetric growth can lead to persisting, relatively large undecorated sections between two slow growing fronts. The process of protein assembly in such regions remains inefficient until a new nucleation event occurs there. In contrast, the prediction of Eq. 9 corresponds to the limit where the number of domains is infinite and therefore, the growth asymmetry has no effect on ϕ ¯ ( t / τ g ) . We also found that the discrepancy between the behavior of ϕ ¯ ( t / τ g ) as obtained from the simulation and that of Eq. 9 diminishes for lower values of k g / k n , due to the larger number of domains present at any particular t / τ g (not shown).

As discussed in the previous section, our experiments were performed in the few nuclei regime where neither Eq. 5 nor Eq. 7 are valid. Instead, we can use the Monte Carlo model described above to predict the expected behavior of ϕ ¯ ( t ) . In Figure 10, we show the kinetics of the RecA decorated fraction on the dsDNA as obtained in five different experiments performed under identical conditions (pH = 7.9, T = 37.0 ± 0.1°C, F = 0.8 ± 0.1 pN, [RecA] = 9.33 μM, [ATPγS] = 4.5 mM, [MgCl₂] = 6.25 mM, [DDT] = 6.25 mM and [TrisHCl] = 18.75 mM). For our analysis, we selected only experiments that are complete, namely, those that reach ϕ ≃ 1 (see Supplementary Section S1). Moreover, since we cannot determine the diffusion time of the proteins between the injection point to the neighborhood of the dsDNA molecule that is being examined, t = 0 was set to the time of the first nucleation event. The stepwise nature of ϕ ( t ) is more pronounced in some of the experiments of Figure 10 (e.g. black and cyan curves) than in others (e.g. blue curve) and the time scale of the kinetics varies significantly between experiments.

The large variability between the experimental kinetics obtained in individual experiments indicates that these cannot be individually described by our Monte Carlo model. Instead, we expect that the model should be able to reproduce the average experimental kinetics, ϕ ¯ ( t ) . Accordingly, we average the time traces of Figure 10 and compare the resulting curve to the prediction of the model. The comparison consists of finding the best fitting theoretical kinetics, ϕ ¯ ( t ) , to the experimental average as a function of the model parameters r , k n , k g ′ (see Figure 11). To obtain the best fit, we minimize the corresponding χ 2 -function which includes the errors of both the theoretical and the experimental ϕ ¯ ( t ) kinetics. These errors are obtained from the fluctuations between the individual time traces, ϕ ( t ) , that are used to compute the average kinetics, ϕ ¯ ( t ) . We find that the value of the χ 2 function manifests a large variability whenever the averaging of the Monte Carlo model is insufficient. To determine the value of the χ 2 function with enough accuracy, ∼1%, we have to use M = 10 5 , mainly in the parameter range around the minimum, which is a relatively heavy computation. Standard fitting routines were unable to obtain the parameter values corresponding to the minimal χ 2 , leading to either local or spurious minima. Instead, our search for the minimum of the χ 2 function is performed by means of a three dimensional scan in the 3 parameter space, r , k n , k g ′ , whereby we average more (larger M) in the regions where the value of χ 2 is smaller. This is an iterative method using first limited averaging to map the r , k n , k g ′ space, M = 10 3 , with a ∼10% accuracy in χ 2 . This is followed by a two step increase in M, M = 10 4 and M = 10 5 , within regions of the parameter space where the χ 2 values obtained in the previous step were below an appropriate threshold. This approach allows us to zoom in on the region of the minimal χ 2 . The step for the scan is chosen such that the error in χ 2 at its minimum, ∼1%, is smaller than the difference between the value of the computed χ 2 at the minimum, χ 2 = 4318 , and that at any of the neighboring points on the scan grid.

For simplicity, we also use the values of the scan steps to represent the errors of the best fitting parameters. These errors are therefore overestimated and should be regarded as upper bounds and of the same order of magnitude as the exact values. We find that the best fit to the experimental ϕ ¯ ( t ) is obtained for r ¯ = 0.10 ± 0.05 , k ¯ n = ( 1.05 ± 0.05 ) ⋅ 10 − 3 ⋅ sec − 1 and k ¯ ⁡ g = ( 5.8 ± 0.3 ) ⋅ 10 − 4 ⋅ sec − 1 . The corresponding values for the average nucleation rate density, n ¯ , and the average domain growth velocity, v ¯ , are n ¯ = ( 9.8 ± 0.5 ) ⋅ 10 − 8 ⋅ binding   site − 1 ⋅ sec − 1 and v ¯ = 6.2 ± 0.3   Re cA ⋅ sec − 1 , similar to the results obtained in the previous section from the analysis of the individual domains (see also Table 1). Due to the variability in the length of the dsDNA’s in our experiments, only the values of the molecular parameters, n ¯ and v ¯ , that were obtained using the methods of this section and those of the previous one, namely, the Monte Carlo model analysis and the domain statistics, respectively, should be equivalent.

Although RecA and dsDNA are complex biomolecules, we may obtain further insight on the way they assemble in the presence of ATPγS by comparing this process to a simple bimolecular reaction of the type A  +  B → C (14) where the Re cA - ATP γ S molecule plays the role of reactant A, the dsDNA is the reactant B and the Re cA - ATP γ S - dsDNA complex represents the corresponding product, C. However, unlike in a standard bimolecular reaction, in our case, one of the reactants, the dsDNA , is an extended filament with multiple binding sites for the Re cA - ATP γ S molecule. Moreover, one can distinguish between two types of binding steps: 1) nucleation type where both neighboring binding sites are unoccupied and 2) growth type where at least one of the two neighboring binding sites is occupied. In what follows, we refer to such reactions as adsorption. In contrast, a simple bimolecular reaction involves two small molecules with a single binding site and its kinetics is determined by the corresponding rate constant, k b .

Despite the apparent differences between the bimolecular adsobtion and reaction, in our case, the two become equivalent in the limit where the dsDNA filament is only three base pairs long, consisting of a single binding site for the Re cA - ATP γ S molecule. In this limit, the reaction kinetics is described by d [ C ] d t = k b [ A ] [ B ] = k b ′ [ B ] = − d [ B ] d t (15) where […] denotes the concentration and we have assumed that [ A ] ≫ [ B ] , such that [ A ] is practically constant and k b ′ = k b [ A ] . Since [ C ] ( t ) + [ B ] ( t ) = [ B ] ( 0 ) , Eq. 15 leads to an exponential growth of the products concentration, [ C ] ( t ) = [ B ] ( 0 ) ( 1 − e − k b ′ t ) . In this case, k b ′ and k n become equivalent.

Since the bimolecular adsorption and simple bimolecular reaction become equivalent in the limit described above, one expects that the exponential behavior of the latter, [ C ] ( t ) = [ B ] ( 0 ) ( 1 − e − k b ′ t ) , can be obtained in a particular regime from the Gaussian kinetics of Eq. 4. For the adsorption, a large number of Re cA - ATP γ S binding sites, L 0 / 3 , are stringed along and the decorated fraction of the dsDNA, ϕ ( t ) , is proportional to the concentration of the products, [ C ] ( t ) . Therefore, there is no limit for which the Gaussian kinetics of Eq. 4 approaches the exponential kinetics of simple bimolecular reaction. However, a closer inspection of Eq. 3 reveals that in Eq. 3b should appear an additional term that accounts for the increase in the occupation of binding sites due to nucleation, k n ( 1 − ϕ ) . On one hand, such term leads to an exponential kinetics for ϕ ( t ) in the limit of small coverage, ϕ → 0 , corresponding to the short time regime before the contribution due to domain growth becomes significant. On the other hand, the k n ( 1 − ϕ ) term is negligible unless the rate of nucleation is extremely large such that the number of nuclei is comparable to the number of sites on the dsDNA filament where nucleation can occur. This is not the case in our experiments where the number of nuclei, N, is typically below 10 while there are on average ( L ¯ 0 / a ) ≅ 11800 binding sites on each dsDNA molecule.

In the case of our adsorption experiments of Re cA - ATP γ S on dsDNA, one cannot expect the value of the average nucleation rate, k n , to be similar to the simple bimolecular reaction rate, k b ′ . The two rates will differ for the following three reasons. First, the value of k n is influenced by the presence of the neighboring sections of the dsDNA on both sides of the binding site, that are absent in the case of the simple bimolecular reaction. Second, in our experiments the dsDNA molecules are tethered to the cover slip at the bottom of the sample, unlike in the simple bimolecular reaction where both reactants are free to diffuse throughout the volume of the sample. This difference between the spatial configurations of the two types of reaction leads to different collision probabilities between the reactant molecules. Finally, the value of k n depends on the number of binding sites on the dsDNA filament, as discussed in relation to Eq. 3, such that it should be compared to k b ′ ( L 0 / a ) rather than k b ′ itself.

Depending on the nature of the reactants, chemical reactions can be either diffusion-limited or reaction-limited. Pugh and Cox have shown that the reaction between RecA and dsDNA is reaction-limited at saturated protein concentration (Pugh and Cox, 1987). Since our experiments are performed in this regime, the corresponding rate constants follow the Arrhenius law, that is, k n ∝ ( L 0 / a ) exp ( − ( E n / k B T ) ) , k g ′ ∝ ( a / L 0 ) exp ( − ( E g f / k B T ) ) and r k g ′ ∝ ( a / L 0 ) exp ( − ( E g s / k B T ) ) , where E n , E g f and E g s denote the activation energies for nucleation, fast front growth and slow front growth, respectively. These relations allow us to obtain the differences between the activation energies from the measured values of the rate constants, namely, Δ E c o o p = E n − E g f = k B T ⁡ ln ( k g ( 1 + r ) k n L 0 2 a 2 ) (16)

and Δ E a s y m = E g s − E g f = − k B T ⁡ ln ( r ) (17)

The value of Δ E c o o p (see Table 1), the decrease in activation energy due to the cooperative interaction between the Re cA - ATP γ S molecules in the fast growing direction, is about 18 k B T . This confirms that protein cooperativity is strong and plays a central role in determining the kinetics of the assembly of Re cA - ATP γ S on dsDNA (Shaner et al., 1987).

Both the L ( t ) decomposition analysis allowing to determine the velocities of the slow and fast front for each individual domain and the comparison between the Monte-Carlo model and the measured average coverage kinetics, ϕ ¯ ( t ) , indicate that the domains grow asymmetrically. That is, domains grow on average about 10 times faster at their fast growing end than at the slow one. Previously, it was shown in biochemical studies that asymmetric assembly of RecA on both ssDNA and dsDNA takes place in the presence of ATP (Register and Griffith, 1985; Shaner et al., 1987; Shaner and Radding, 1987). Moreover, it was found that the fast growth is oriented in the 5′ to 3′ direction. It was suggested that the asymmetric growth of the RecA domains is due to the hydrolysis of the ATP in the RecA-ATP-DNA complex (Cox, 2007a)^. More recently, Galletto et al. used fluorescently labeled RecA to image the kinetics of domain growth on individual dsDNA molecules (Galletto et al., 2006). They found that while RecA domain growth is slower in the presence ATP γ S than in the case when ATP is used, the domains grow asymmetrically for both. Although Galletto et al. (Galletto et al., 2006) have characterized the extent of the asymmetry in the domain growth on a mainly qualitative level, for some of their experiments with ATP γ S (at particular NaCl concentrations) domain growth appears to be almost unidirectional [see Figure 4B in (Galletto et al., 2006)]. Such behavior is consistent with the behavior found in our experiments where the slow growth front of Re cA - ATP γ S - dsDNA is hardly advancing relative to the fast front.

Unlike in the biochemical studies, neither in our experiments nor in those of Galletto et al. (Galletto et al., 2006) can one establish the relation between the domain growth asymmetry and the direction along the DNA. However, we can use Eq. 17 to determine the difference between the energy barriers for Re cA - ATP γ S binding at the slow and fast ends of a domain. As was discussed above, the value of r obtained from the fit of the ϕ ¯ ( t ) computed from the Monte-Carlo model to the one measured in experiments, r ¯ = 0.10 ± 0.05 , is significantly more reliable than the value found by averaging the growth velocity ratios of individual domains, r ¯ = 0.25 ± 0.08 . Therefore, we expect that the corresponding value of Δ E a s y m (see Eq. 17 and Table 1), Δ E a s y m = ( 2 . 3 ± 0 . 5 ) k B T , represents a good estimate to the actual difference between the energy barriers for Re cA - ATP γ S binding at the slow and fast ends of the Re cA - ATP γ S - dsDNA domain.

Regarding the mechanism of asymmetric domain growth of Re cA - ATP γ S on dsDNA, it may be understood assuming that the Re cA - ATP γ S complex undergoes a conformational change following its binding to the dsDNA (see Figure 12). Specifically, we propose that the conformation of the Re cA - ATP γ S complex is such that it is much more likely to bind at one end of the Re cA - ATP γ S - dsDNA domain than at the other. After binding however, the conformation of the Re cA - ATP γ S complex changes such as to allow the next Re cA - ATP γ S complex to bind to it. Previously, similar mechanisms were proposed to describe the treadmilling of actin filaments (Neuhaus et al., 1983). To establish the validity of the scenario depicted in Figure 12 detailed information on the structural differences between the free and the dsDNA bound Re cA - ATP γ S is necessary. Such analysis is beyond the scope of this study.