We calculated the first-order reaction rate constant k 2 as follows. Throughout this discussion, T 2 represents freely diffusing transposase, which is shown as T 2,free in the model diagram.

Consider the two elementary reactions from our model that are shown at the bottom-left and the top-right regions of Figure 1I, respectively. Ignoring the transposon’s internal DNA sequences for now, these reactions are (1) T 2 + B → BT 2 and (2) AT 2 + B → AT 2 B. Both reactions involve the binding of transposase to DNA site B, so they are chemically identical. This might suggest that they would have the same reaction rates. However, in actuality, their reaction rates differ because the rate of diffusion for the free transposase is different from that of the DNA-bound transposase. Expressing their reaction rate constants as k 1 for reaction 1 (as in the model diagram) and k 1 ′ for reaction 2, the respective reaction rates are

(1) d [ BT 2 ] dt = k 1 [ T 2 ] [ B ] ,

(2) d [ AT 2 B ] dt = k 1 ′ [ AT 2 ] [ B ] .

The Collins and Kimball reaction rate theory (Rice, 1985) enables us to expand these reaction rate constants into one contribution that arises from diffusion and a second that arises from the binding activation energy. According to this theory, the expanded reaction rate constants are

(3) 1 k 1 = 1 k 1 , diff + 1 k act ,

(4) 1 k 1 ′ = 1 k 1 , diff ′ + 1 k act ,

where k 1,diff and k 1 , diff ′ are diffusion-limited reaction rate constants and k act is an activation-limited reaction rate constant. These equations include the same k act value because we assumed above that the two reactions are chemically identical; under this assumption, they should have the same binding activation energy and thus the same activation-limited reaction rate constants.

The necessary diffusion-limited reaction rate constants can be calculated from equations that Berg derived for the diffusion-limited association rates of proteins to DNA (Berg, 1984). Berg’s equation 16 is for the association rate of a freely diffusing protein to a specific DNA site, while his equation 27 is for the association rate of a DNA-bound protein to another specific DNA site. In our notation, these equations are

(5) k 1 , diff = 4 π D p R + D s a ( R a ) 1 / 3 ,

(6) k 1 , diff ′ = 1.4 D s a ( R a ) 1 / 3 .

These equations were derived by treating DNA with the worm-like chain model, in which DNA is represented as a long thin semi-flexible filament that bends smoothly over the course of its length. This model was recently shown to be reasonably accurate for modeling DNA dynamics (Petrov et al., 2006). The central model parameter is the filament persistence length, a, which is the characteristic length for filament bending. According to the model, DNA binding sites diffuse rapidly within their local regions, while also gradually diffusing away to more distant regions. The rate of this local ‘segmental diffusion’ is quantified with the diffusion coefficient D s . D s is the translational diffusion coefficient of a free DNA fragment with length a (Berg, 1984). Continuing with the above equations, R represents the distance over which a specific interaction occurs between transposase and its DNA binding sites. It arises from the assumption that the two reactants (the transposase binding domain and a DNA binding site, in this case) can be treated as hard spheres that react upon collision. Finally, D p is the diffusion coefficient of free transposase relative to the center of mass of the DNA.

Two final equations need to be introduced to enable us to calculate the rate of synapsis, k 2 . The reaction rate equation for reaction 2, given in equation 2, can be rearranged by grouping k 1 ′ and [B],

(7) d [ AT 2 B ] dt = ( k 1 ′ [ B ] ) [ AT 2 ] = k 2 [ AT 2 ] .

The latter equality defines k 2 , a pseudo-first order reaction rate constant, as k 1 ′ [ B ] . Here, [B] is the concentration of the B DNA binding site in the vicinity of the A binding site. We used the following empirical equation from Ringrose et al. (1999) to find [B]:

(8) j M = ( 4 a 10 4 b ) 3 / 2 exp ( − 460 a 2 6.25 b 2 ) ( 1.25 ⋅ 10 5 a 3 ) .

j M is the local concentration of one DNA site in the vicinity of another in M, a is the DNA persistence length in nanometers, and b is the separation between the sites in base pairs.

Table 2 lists the values that we used and derived to estimate k 2 . In brief, we used equation 3, including an experimental value for k 1 and a calculated value for k 1,diff from equation 5, to calculate k act . We then substituted k act into equation 4, along with a calculated value for k 1 , diff ′ from equation 6, to estimate k 1 ′ . Finally, we used equations 7 and 8 to estimate k 2 , the rate of synapsis, from k 1 ′ .

These theoretical considerations allow us to develop a model of an idealized transposition reaction, in which the monomers within a transposase dimer bind the first and second transposon ends with equal affinity. In this idealized situation, synapsis of the transposon ends is simply a product of the sequential binding of the transposase dimer to sites at opposite ends of the element. We then go on to consider a more realistic model in which allosteric interactions between the subunits reduces the affinity of the developing transpososome for the second transposon end. The magnitude of this effect is provided by experimental estimates of the rate of synapsis (Figure 3D, Table 2; Claeys Bouuaert and Chalmers, 2010; Claeys Bouuaert et al., 2011).

In Table 2, the reaction radius, R, is a fairly rough estimate. However, this is not a major concern because reaction rates are relatively insensitive to the reaction radius (Berg, 1984) (they scale as R1/3). Other values are likely to be more accurate. A result of the calculations shown there is that k 1 ′ is only about a factor of 1.6 slower than k 1 . This indicates that the slower diffusion of DNA-bound transposase than of free transposase has only modest impact for in vitro experiments.

For Tn10/5 there is no experimental data regarding the affinity of the transposon-end-bound monomers when they collide with each other and achieve synapsis (see kinetic model in Figure 1B). For the purposes of the simulation we assigned an association rate constant equal to that for the mariner transposase binding to a transposon end. In the respective prokaryotic and eukaryotic models, the rate of synapsis is therefore determined by the same association rate constant and the rate of diffusion, which is identical in both systems. The outputs of the respective simulations are therefore directly comparable and reveal differences in the underlying kinetic models.