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Comparison of kinetic and dynamical models of DNA-protein interaction and facilitated diffusion

It has long been asserted that proteins like transcription factors may locate their target in DNA sequences at rates that surpass by several orders of magnitude the three-dimensional diffusion limit thank to facilitated diffusion, that is the combination of one-dimensional (sliding along the DNA) and three-dimensional diffusion. This claim has been supported along the years by several mass action kinetic models, while the dynamical model we proposed recently (J. Chem. Phys. 130, 015103 (2009)) suggests that acceleration of targeting due to facilitated diffusion cannot be large. In order to solve this apparent contradiction, we performed additional simulations to compare the results obtained with our model to those obtained with the kinetic model of Klenin et al (Phys. Rev. Letters 96, 018104 (2006)). We show in this paper that the two models actually support each other and agree in predicting a low efficiency for facilitated diffusion. Extrapolation of these results to real systems even indicates that facilitated diffusion necessarily slows down the targeting process compared to three-dimensional diffusion.


Introduction
Prof. S.E. Halford (University of Bristol, UK) recently published an article entitled "An end to 40 years of mistakes in DNA-protein association kinetics ?" [1]. In this article, he fights against the popular belief that some proteins, like transcription factors, can bind to their specific DNA targets at rates that surpass the diffusion limit, that is the rate at which the protein and the DNA collide with each other as a result of 3-dimensional (3D) thermal diffusion. According to him, there is "no known example of a protein binding to a specific DNA site at a rate above the diffusion limit" [1].
The whole story actually started with the pioneering work of Riggs, Bourgeois and Cohn [2], which was completed shortly after the first demonstration that proteins can bind to specific DNA sequences [3,4]. Riggs et al reported that, in a buffer containing KCl, Tris-HCl and magnesium acetate at 0.01 M concentrations (M means mole per liter), the Lac repressor binds to its operator site at a rate of about 9 10 7 × M -1 s -1 [5]. This value is one to two orders of magnitude larger than the one that is generally assumed for the diffusion limit of protein-DNA association, that is 10 8 M -1 s -1 [5,6].
Elaborating on the early remarks of Adam and Delbrück [7] and Richter and Eigen [8], that reducing the dimensionality of diffusion-based reactions can greatly increase their efficiency, Berg and coworkers concluded that the result of Riggs et al proves that DNAprotein interactions cannot rely on pure 3D diffusion, but must instead involve alternate sequences of 3D diffusion of the protein in the buffer and 1-dimensional (1D) sliding of the protein along the DNA sequence. This process is known as "facilitated diffusion". Most mass action kinetic models developed today to investigate facilitated diffusion are still based on the model that Berg and coworkers proposed to legitimate their claim [9][10][11][12].
The assertion that facilitated diffusion may greatly enhance DNA-protein association rates -or equivalently that the rate measured by Riggs et al is much larger than the diffusion limit -is instead more questionable. It was precisely the purpose of Ref. [1]   is of the order of 0.5 nm, that is 6 We recently proposed a "dynamical" (or "molecular mechanical") model for nonspecific DNA-protein interactions and facilitated diffusion, that is a microscopic model which relies uniquely on the definition of a Hamiltonian describing all interactions inside the model cell and the choice of equations of the motion [26,27]. This model is based on the "worm-like chain and beads" or "bead-spring" model for DNA, which was developed by several groups [28][29][30][31][32][33]. We added to this DNA model one or several interconnected beads to describe the protein and paid special attention to the terms of the Hamiltonian that model the interactions between DNA and the protein [26,27]. We used the Brownian dynamics algorithm of Ermak and McCammon [34], which includes hydrodynamic interactions, to propagate classical trajectories. This confirmed that our model satisfactorily reproduces the succession of motions by which the protein samples the DNA sequence, that is 3D diffusion in the buffer, 1D sliding along the sequence, short hops between neighboring sites, long hops between more distant sites, and intersegmental transfers. We also showed that the portion of time during which the protein slides along the DNA can be varied from nearly 0 to 1 by increasing the charge(s) placed at the center of the protein bead(s), which provides a straightforward method to investigate quite different dynamical regimes. One of the main results of [26,27] is that this model suggests that, whatever the portion of time the protein spends in 1D sliding along DNA, the facilitated diffusion mechanism cannot increase the sampling speed by more than 70% when the protein is described by a single bead [26]. The maximum speed increase may even be much smallerand vanish -when the protein is modelled as a set of interconnected beads [27]. Results obtained with this model therefore agree with the claim of Prof. Halford [1].
It seems, however, that the results obtained with our dynamical model differ from those obtained with mass action kinetic models, that is models based on an a priori scenario of site targeting and the definition of a more or less extended list of phenomenological rates and probabilities. These later models indeed usually predict that the facilitated diffusion mechanism is capable of increasing the targeting speed by one to several orders of magnitude [35][36][37][38][39]. We therefore tried to compare our model to kinetic ones, in order to understand what this difference is due to. It rapidly appeared that several of the kinetic models more or less explicitly disregard exact numerical coefficients. For example, [35,36] neglect the 4π coefficient in Eq. (1.1). As a result, these models provide simple analytical results that are very useful to decipher the scaling dependence of the targeting speed on major system parameters, but they cannot provide reliable estimates of the speed increase due to facilitated diffusion. Other models rely heavily on terms or expressions, which are quite difficult to relate to the properties of real molecules and/or quantities derived from dynamical systems (see for example the last term in the right-hand side of Eq. (6) of [39]).
We essentially found only one kinetic model to which to compare the results obtained with our dynamical model. Using older calculations of Szabo et al [40], Klenin, Merlitz, Langowski and Wu indeed obtained that the mean time of the first arrival of the protein at the target of radius a can be written in the form [38] )] arctan( where 1D D and 3D D are the diffusion coefficients of the protein in the buffer and along the DNA sequence, respectively, V is the volume of the buffer, L the length of the DNA sequence, and ξ the distance τ and 3D τ are the average times the protein spends in the bound and free states, respectively). The accuracy of Eq. (1.3) was checked for the simple system where the protein is described as a random walker that is allowed to enter freely in the neighbourhood of the DNA but has a given finite probability to exit this volume at each time step [38,41,42].
As will be shown below, all the quantities that appear in

Dynamical model and Brownian dynamics simulations
As illustrated in Fig. 1 placed at its center ( e is the absolute charge of the electron). In [26,27], the radius 0 R of the sphere was chosen so that the density of bases inside the cell is close to the physiological value. In the present work, we instead varied the DNA concentration, in order to better check the respective influences of 1D and 3D motions (see below). As pointed out in [35], the volume V of the buffer is connected to the total DNA length L according to   [26], respectively. It is important to realize that the interaction energy between a DNA bead and the protein one must be minimum for a distance close to the sum of their radii, 28 . 5 prot DNA = + = a a σ nm, in order for 1D sliding to take place. The expression for the excluded volume term ev E in eq (2.6) of [26] insures that this is indeed the case and that the position of the minimum does not depend on the protein charge prot e . As will be discussed in some detail below, we also considered the case where the DNA-protein interaction is purely repulsive (see Fig. 1 of [26]). The repulsive potential is obtained by keeping only the repulsive part of the interaction potential for , that is by shifting the energies so that the minimum energy is zero and setting to zero the interaction energy for DNA-protein distances larger than the position of the minimum.
We performed two different sets of simulations. For the first set, the effect of the buffer was modeled in the simple standard way by introducing a dissipative and a stochastic term in the equations of motion. For the second set of simulations, we additionally took hydrodynamic interactions (HI) into account, that is the influence of the motion of a given bead on the motion of the neighboring ones. For simulations where HI were neglected, the positions of all of the beads were updated at each time step according to Eq. (2.10) of [26], which is just the discretization of standard Langevin equations without inertial contributions. For simulations which took HI into account, the positions of the protein bead and the 100 DNA beads closest to it were instead updated according to the BD algorithm of Ermak and McCammon [34], that is Eq. (2.8) of [26]. Since we are essentially interested in the time evolution of the number of different DNA beads visited by the protein, the results obtained with this procedure are identical to those obtained by using the algorithm of Ermak and McCammon to update the positions of all of the 4000 DNA beads, while the required CPU time is orders of magnitude times shorter [26].
For the sake of completeness, let us finally note that all simulations were performed with a time step 400 = ∆t ps and began with a 40 µs thermalization cycle. Moreover, all the results presented below correspond to the average of at least four simulations with different random initial conditions.

Method of comparison
In this section, we describe the method we used to compare our model [26] with that of Klenin et al [38].
The basic information, which we extract from Brownian dynamics simulations, is the time evolution of the number ) (t N of different DNA beads visited by the protein. As will be illustrated and discussed in more detail below, we found that, for all the simulations we ran, ) (t N follows very precisely the same law as we already observed in our previous studies [26,27], that is is the total number of DNA beads. By inverting this relation, one obtains that the time k t of first arrival at the kth distinct bead is This relation is, however, necessarily wrong for the last DNA bead ( n m k = ), since it predicts that it takes an infinite time for the protein to reach this bead, while this time must This means that the number of different DNA beads visited by the protein increases linearly with time as long as saturation does not set in. At first, we believed that this indicates that the global 3D motion of the protein is not diffusive [26]. However, we then realized that, according to a rather old mathematical result known as the "volume of the Wiener sausage" [43], the volume linearly with time according to [44,45] where δ is the maximum distance away from the central Brownian motion of each point of the solid. If one assumes that DNA is homogeneously distributed in the cell and that the diffusive motion of DNA is slow compared to that of the protein, then

Results
In this section, we report the values of the rate constant κ, the fraction of time D  ), and two different ways of handling HI ("off" and "on"). As will also be the case for all subsequent tables, the first number in each entry was obtained with the where D 1 ρ denotes the fraction of time during which the protein is attached to a DNA bead, Values of D 1 ρ are easily extracted from the simulations by checking at each time step whether the distance between the center of the protein bead and that of any DNA bead is smaller than the threshold, that is σ or 1.5σ. The obtained values of D 1 ρ are shown in Table 2. As already emphasized in [26,27], D 1 ρ increases from nearly 0 for the repulsive potential to almost 1 for large values of the protein charge. It can also be seen in Table 2 that D   1 ρ is substantially smaller when HI are taken into account than when they are not. Stated in other word, HI tend to move the protein away from the DNA. We will see that this has a marked effect on the targeting speed. At last, it can also be noticed ) during more than 0.07 µs, (iii) the protein bead did not reach one of the extremities of the DNA segment. We then computed the average value of ) (t N during these events and drew log-log plots of the time evolution of ) (t N . A few representative plots are shown in Fig. 3 is the length of the DNA sequence visited by the protein after time t. One therefore obtains Values of 1D D obtained with this method are reported in Table 3 Table 4. It can be checked that they indeed agree very closely with the result of Einstein's formula. In contrast, when HI are taken into account, the values of 3D D obtained from the expression of the volume of the 3D Wiener sausage in Eq. (3.7) are substantially larger than those obtained from Einstein's formula (see the bottom half of Table 4). This increase of 3D D when HI are turned "on" was rather unexpected to us, because it has long been known that HI tend to decrease the association rate between two diffusing spheres placed at short distance [47][48][49]. This is due to the fact that the stochastic (thermal) motions of the two particles become highly correlated, so that their relative mobility slows down. But, on the other hand, this increase of 3D D agrees with Kirkwood-Riseman's equation, which states that HI reduce the effective friction coefficient of long DNA chains [50]. Anyway, when estimating 3D D in the standard way, by fitting the mean squared displacement of the protein with a linear law, that is we again obtained that 3D D is larger when HI are turned "on" than "off". For example, we checked that Eq. ρ is actually small but not zero because of collisions (see Table 2). As a consequence, the "repulsive potential" column of Table 5 is similar to that of

Discussion
In this section, we first discuss the results obtained with the dynamical model and then the extent to which these results agree with those of the kinetic model of Klenin et al [38]. We postpone the discussion of the conclusions concerning real systems, which can be drawn from this comparison, to the next, conclusive section. Table 6 shows the acceleration of the protein targeting process due to facilitated diffusion. This acceleration was estimated as the ratio of a given rate constant κ for  Table 7 similarly shows the acceleration of the targeting process due to HI. This acceleration was estimated as the ratio of a given rate constant κ for HI switched "on" divided by the corresponding value of κ for HI switched "off". In both cases, the values of κ were taken from Table 1 for the dynamical model and from Table 5 for the kinetic model.

A -Acceleration of targeting due to facilitated diffusion and hydrodynamic interactions
Let us first concentrate on the results obtained with the dynamical model. For a reason, which will become clear later, we discuss in the present subsection only the results obtained with the σ threshold. For HI switched "on", the values for the acceleration of targeting due to facilitated diffusion reported in Table 6 just match those obtained in our first study [26]. First, as already noted in the preceding section, both 1D D and 3D D are roughly twice larger when HI are switched "on" than when they are switched "off" (see Tables 3 and 4). This, of course, accelerates the targeting process in proportion. The second effect is that HI tend to detach the protein from the DNA sequence, as can be checked by looking at the values of D 1 ρ reported in Table 2. This considerably modifies the motion of highly charged proteins.

B -Comparison of the dynamical and kinetic models
Let us now examine the degree of agreement between results obtained with the dynamical and kinetic models, and let us start with the results obtained when switching HI "off". For the repulsive DNA/protein interaction potential, the corresponding columns of Tables 1 and 5 are identical. This actually just reflects the facts that the values of κ in Table   1 were used to estimate the diffusion coefficients 3D D reported in Table 4 and that D 1 ρ was further assumed to be zero in Eq.  Table 4, one again obtains "kinetic" rate constants κ that are in excellent agreement with "dynamical" ones. The top half of the last column of Table 5 is void. This is due to the fact (already ρ deduced from simulations with HI switched "on" to get reasonable estimates of κ ?
Comparison of the bottom halves of Tables 1 and 5 suggests that this is indeed the case.
Even if the values of κ differ in one case by a factor of 2, the agreement is generally correct.

Conclusion
In this work, we have thus shown that the dynamical model we proposed [26,27] and the kinetic model of Klenin et al [38] support each other, in the sense that the rate constants κ obtained (i) directly from the simulations, and (ii) from Klenin et al's formula using values of 1D D , 3D D , and D 1 ρ extracted from the simulations, are in good agreement.
In particular, both models suggest that the acceleration of targeting due to facilitated diffusion is not very large for the system we considered. Table 6 indeed shows that the dynamical and kinetic models agree in predicting an acceleration comprised between 20% and 70% for physiological values of w, HI switched "on", and protein charges ranging from Quite obviously, one must at this point wonder how this result transfers to real DNA and proteins. The essential point is that the dynamical system corresponds to a ratio 3D 1D / D D of the order of unity (see Tables 3 and 4), as is customary for translational diffusion. In contrast, the ratio 3D 1D / D D for real DNA/protein systems (measured essentially by single molecule experiments) is rather of the order of 3 10 − ≈ [13,14,16,[18][19][20][21][51][52][53]. This three orders of magnitude difference may be due to the fact that in real systems the protein has to follow an helical track along the DNA, which considerably enhances the translational friction coefficient [15,54,55].  Tables 3 and 4), Fig.   4 indicates that maximum acceleration due to facilitated diffusion is of the order of 2 for physiological values of w, which is exactly what we obtained (see Table 6). In contrast, This conclusion agrees with experimental results, which indicate that the measured apparent diffusion coefficient of molecules that do not interact with chromatin or nuclear structures (like the green fluorescent protein or dextrans) range between 10 -11 and 10 -10 m 2 s -1 [51][52][53], depending on their size, as predicted by Einstein's formula, while that of biologically active molecules is instead usually reduced by a factor of 10-100 compared to this formula [56][57][58][59][60].
In conclusion, we have shown that even in the favorable case studied here  [61,62] and short-range lubrication effects [63], are disregarded.
Still, we cannot think of physical effects that would invalidate the essential conclusions drawn in this work. We consequently agree with Prof. S.E. Halford, that it would now be time to put "an end to 40 years of mistakes in DNA-protein association kinetics".

APPENDIX A : THE DEBYE-SMOLUCHOWSKI'S RATE
If the salinity of the buffer is very low, it can be considered at first order that the electrostatic interaction between DNA and the protein is an unscreened Coulomb potential ( ) , where A q and B q are the charges on interacting particles A and B, and r is the distance between them. Debye [64] showed that, in this case, the association rate is and an excluded volume term, as we used in our simulations [26]. It was however checked numerically that the association rate for a screened Debye-Hückel potential is comprised between Schmolukowski's rate and Debye's one [67].     (3.7) and the values of κ reported in the "repulsive potential" column of Table 1. ρ is actually small but not zero (see Table 2). Moreover, for 3 / DNA prot = e e and hydrodynamic interactions switched "off", the sliding motion of the protein along the DNA is too subdiffusive to enable an estimation of 1D D . Klenin et al's formula can therefore not be used in this latter case.   Table 1 for the   dynamical model and from Table 5 for the kinetic model. Acceleration of targeting due to HI was estimated as the ratio of a given rate constant κ for HI switched "on" divided by the corresponding value of κ for HI switched "off". In both cases, the values of κ were taken from Table 1 for the dynamical model and from Table 5 for the kinetic model.   averaged over several tens of sliding events with the following properties : (i) each sliding event lasted more than 1 µs, (ii) the protein did not separate from the DNA segment by more than σ during more than 0.07 µs, (iii) the protein bead did not reach one of the extremities of the DNA segment.