Facilitated Dissociation Kinetics of Dimeric Nucleoid-Associated Proteins Follow a Universal Curve

Abstract

Recent experimental work has demonstrated facilitated dissociation of certain nucleoid-associated proteins that exhibit an unbinding rate that depends on the concentration of freely diffusing proteins or DNA in solution. This concentration dependence arises due to binding competition with these other proteins or DNA. The identity of the binding competitor leads to different qualitative trends, motivating an investigation to understand observed differences in facilitated dissociation. We use a coarse-grained simulation that takes into account the dimeric nature of many nucleoid-associated proteins by allowing an intermediate binding state. The addition of this partially bound state allows the protein to be unbound, partially bound, or fully bound to a DNA strand, leaving opportunities for other molecules in solution to participate in the unbinding mechanism. Previous models postulated symmetric binding energies for each state of the coarse-grained protein corresponding to the symmetry of the dimeric protein; this model relaxes this assumption by assigning different energies for the different steps in the unbinding process. Allowing different unbinding energies not only has equilibrium effects on the system, but kinetic effects as well. We were able to reproduce the unbinding trends seen experimentally for both DNA and protein competitors. All trends collapse to a universal curve regardless of the unbinding energies used or the identity of the dissociation facilitator, suggesting that facilitated dissociation can be described with a single set of scaling parameters that are related to the energy landscape and geometric nature of the competitors.

Introduction

Physical interactions between proteins and DNA are an important and persistent area of scientific interest (1, 2, 3, 4). A full understanding of these behaviors requires an accounting of how DNA and proteins work together or compete at a molecular level. These connections govern cellular processes, such as mRNA transcription (4, 5, 6, 7, 8, 9), gene expression (4, 10, 11, 12), and cell division (4, 13). An important example of this is the interplay between genomic DNA and nucleoid-associated proteins (NAPs) in prokaryotes (14, 15, 16, 17, 18, 19). Increased understanding about the role of proteins in the nucleoid has led to an appreciation of the wide variety of behavior, both physical and biological, among NAPs (4, 11, 20). NAPs are required for cell processes to occur, often relying on their ability to manipulate nucleoid architecture; however, the exact mechanisms involved in these processes are still not fully understood (4, 21, 22, 23, 24).

The properties of the bound NAP-DNA complex has been extensively studied, due to the importance of NAP-DNA interactions (25, 26, 27, 28, 29, 30, 31, 32) and binding kinetics (19, 33) in determining cell physiology. Experimental work has demonstrated that once NAPs are bound to DNA, they are typically stable and stay bound for long periods of time in the absence of other molecules, such as freely diffusing proteins or protein-free DNA (29, 30, 34). The presence of these molecules, however, has a pronounced effect on NAP-DNA dissociation kinetics. Recent single-molecule measurements, using force spectroscopy and fluorescence microscopy, have determined the unbinding kinetics of a variety of NAPs, including FIS and HU in the presence of other NAPs in solution (34, 35). These molecules are thought to facilitate the dissociation of the proteins bound to the DNA by competing for binding locations. Initial studies with competing NAPs in solution unexpectedly found a linear concentration-dependent unbinding rate (35):

$$k_{\text{off}}=k_0+k_{1}c.$$ (1)

Here, $k_0$ is the unbinding rate constant in the limit of zero concentration and $k_1$ is the unbinding coefficient that incorporates the linear protein concentration c dependence of the unbinding rate constant.

Further studies incorporated an external DNA concentration instead of an external protein concentration (34). These studies showed a different concentration-dependent unbinding rate,

$$k_{\text{off}}=\frac{c}{Ac+B},$$ (2)

where A and B are constants that can be fit experimentally, and c is the freely diffusing DNA concentration. The off-rate increases and then plateaus, indicating that the addition of any more competitor (DNA) will not further facilitate the dissociation of the bound NAPs.

Facilitated dissociation mechanisms, both general and system-specific, have been proposed to describe these different behaviors (34, 36, 37, 38, 39, 40, 41, 42, 43). Some mechanisms predict a purely linear relationship between the competitor concentration and the $k_{\text{off}}$ (39), while others predict that $k_{\text{off}}$ will plateau at high competitor concentrations (34, 36, 42). Mechanisms that describe the plateau behavior include two limiting behaviors: an initial linearly dependent increase in the off-rate, then leveling off to a concentration-independent (or pseudo-saturated) off-rate (34). Recent work from Giuntoli et al. (34) provides a hypothesized model for dimeric NAP dissociation facilitated by DNA strands in solution. Their model is capable of capturing the general trends seen experimentally, but has not been directly related to the binding and unbinding energy landscape.

The NAP FIS has been shown to demonstrate both linear and plateau behavior depending on the nature of the competitor. Motivated by the FIS-DNA system, the intent of this work is to build upon previous protein-DNA binding and unbinding models to capture both the linear and plateau unbinding rates of FIS from DNA (34, 35, 39). Our simulations show that both experimentally observed trends arise from the same general model, and are part of a larger continuum among facilitated, nonfacilitated, and maximally facilitated dissociation behaviors. We expect that our work has broader implications than just the FIS-DNA system, which may be extended to NAP-DNA or other biomolecular systems that undergo facilitated dissociation (23, 44, 45, 46).

Materials and Methods

To investigate the binding and unbinding kinetics of dimeric NAPs from DNA, we use a coarse-grained Brownian dynamics model that incorporates NAP dimeric structure and a proposed binding energy landscape. The simulations are designed such that they mimic the single-molecule experiments performed by Graham et al. (35) and Giuntoli et al. (34), and also build on previously proposed models of the system (39). We initialize our system with a static linear chain of N beads of radius $a=2.0$ nm to represent the tethered DNA strand. FIS proteins are modeled as $n_D$ dimers of beads, and freely diffusing DNA strands are modeled as $n_s$ chains of M beads. The simulation box size is 200a × 200a × 224a with periodic boundary conditions. Simulation snapshots are shown in Fig. 1 A.

Figure 1.

(A) Snapshots from the simulations. Tagged proteins are colored light blue, untagged or nontagged proteins are dark blue, and DNA is yellow. (B) Energy landscape of the system, demonstrating relevant variables. Three states are present: unbound (left), singly bound, and doubly bound (right). $\Delta {\tilde{E}}_1$ is the energy difference between unbound and singly bound, and $\Delta {\tilde{E}}_2$ is the energy difference between singly and doubly bound. $\Delta {\tilde{E}}_{\text{total}}=\Delta {\tilde{E}}_1+\Delta {\tilde{E}}_2$, and $\Delta {\tilde{E}}_i=\Delta {\tilde{E}}_{i,UB}-\Delta {\tilde{E}}_B$. To see this figure in color, go online.

The movement of individual beads of index i is governed by the Langevin equation:

$$\frac{\partial {\mathbf{r}}_i}{\partial t}=-\sum _j^{N_{\text{total}}}{\mu }_{\mathbf{i}\mathbf{j}}{\nabla }_{j}U\left(t\right)+{\xi }_i,$$ (3)

where each particle i has a radius of a and position ${\mathbf{r}}_{\mathbf{i}}$. The value ${\mu }_{\mathbf{i}\mathbf{j}}={\delta }_{ij}\delta /\left(6\pi \eta a\right)$ is the freely draining Stokes mobility matrix, η is the solvent viscosity, δ is the identity matrix, ${\xi }_i$ is a random velocity that satisfies $\langle {\xi }_i{\xi }_j\rangle =2k_{\text{B}}T{\mu }_{\mathbf{i}\mathbf{j}}{\delta }_{ij}$, and ${\delta }_{ij}$ is the Kronecker delta. The energy of the system is normalized by $k_{\text{B}}T$ $\left(\tilde{U}=U/k_{\text{B}}T\right)$, distances are normalized by a $\left({\tilde{r}}_{ij}=r_{ij}/a\right)$, and time is normalized by the diffusion time of a single bead ${\tau }_D=6\pi \eta a^3/\left(k_{B}T\right)$. A tilde indicates that a value is normalized and dimensionless.

The potential energy of the system, U, is the sum of three contributions, $U=U_S+U_{LJ}+U_B$, representing connectivity, excluded volume, and bending, respectively. $U_S$ is a harmonic spring potential:

$${\tilde{U}}_S=\frac{{\tilde{\kappa }}_s}{2}\sum _{ij}{\left({\tilde{r}}_{ij}-{\tilde{r}}_0\right)}^2,$$ (4)

where ${\tilde{r}}_{ij}$ is the distance between the two beads i and j, and the spring constant is set to ${\tilde{\kappa }}_s=200.0$ to prevent large deviations from the equilibrium bond length, ${\tilde{r}}_0=2.0$. Beads in the protein dimers are connected via this harmonic potential, as are the DNA beads in the freely diffusing strands.

The repulsive part of a Lennard-Jones (LJ) potential, $U_{LJ}$ accounts for excluded volume interactions between beads:

$$\{\begin{array}{cc}{\tilde{U}}_{LJ}=\widetilde{\mathit{ϵ}}\sum _{ij}\left[{\left(\frac{{\tilde{r}}_0}{{\tilde{r}}_{ij}}\right)}^{12}-2{\left(\frac{{\tilde{r}}_0}{{\tilde{r}}_{ij}}\right)}^6+1\right]& {\tilde{r}}_{ij}\le 2.0\\ 0& {\tilde{r}}_{ij}>2.0\end{array},$$ (5)

where the LJ parameter $\widetilde{\mathit{ϵ}}=0.41$ sets the magnitude of the excluded volume potential.

The bending potential, $U_B$, is included for freely diffusing DNA strands:

$${\tilde{U}}_B=\frac{{\tilde{\kappa }}_{\theta }}{2}\sum _{i=1}^{m-1}{\left({\theta }_i-{\theta }_0\right)}^2,$$ (6)

where ${\theta }_i$ is the angle between the bonds of beads $i-1$ and i, and i and $i+1$. ${\theta }_0$ is the equilibrium angle between the bonds of beads $i-1$ and i, and i and $i+1$ and is set to ${\theta }_0=0.0$. The bending force constant is set at ${\tilde{\kappa }}_{\theta }=12.5$, which is chosen to match the dsDNA persistence length of 50 nm.

The binding energy landscape of the system is controlled by the binding and unbinding barriers, shown in Fig. 1 B. Each bead in the protein dimer can independently bind (and unbind) to DNA, and the protein can therefore be in one of three states, shown in Fig. 1 B. The dimer can be completely unbound from the DNA (UB), bound by a single bead to the DNA (SB), or bound by both beads in the dimer to the DNA (DB). Each DNA bead can only bind to a single protein bead, so doubly bind proteins occupy two adjacent DNA beads. The binding and unbinding occur via a Monte Carlo update step occurring every ${\tilde{\tau }}_0=0.05$, which demonstrates the statistical results expected from a Bell model type reaction (47). The connectivity of a protein bead (i) and DNA bead (j) is modeled as a harmonic potential that is recalculated every ${\tilde{\tau }}_0$ as follows:

$${\tilde{U}}_C=\frac{{\tilde{\kappa }}_s}{2}\sum _{ij}{\omega }_{ij}{\left({\tilde{r}}_{ij}-{\tilde{r}}_0\right)}^2,$$ (7)

$${\omega }_{ij}\left(t\right)=\{\begin{array}{l}\{\begin{array}{c}1\text{if}\text{Ξ}<e^{-\Delta E_B}\\ 0\text{if}\text{Ξ}>e^{-\Delta E_B}\end{array}\text{if}\sum _k\left({\omega }_{kj}\left(\tilde{t}-{\tilde{\tau }}_0\right)+{\omega }_{ik}\left(\tilde{t}-{\tilde{\tau }}_0\right)\right)=0\text{and}{\tilde{r}}_{ij}<2.1\\ \{\begin{array}{c}1\text{if}\text{Ξ}<e^{-\Delta E_{\text{UB},2}}\\ 0\text{if}\text{Ξ}>e^{-\Delta E_{\text{UB},2}}\end{array}\text{if}{\omega }_{ij}\left(\tilde{t}-{\tilde{\tau }}_0\right)=1\text{and}\text{DB}\\ \{\begin{array}{c}1\text{if}\text{Ξ}<e^{-\Delta E_{\text{UB},1}}\\ 0\text{if}\text{Ξ}>e^{-\Delta E_{\text{UB},1}}\end{array}\text{if}{\omega }_{ij}\left(\tilde{t}-{\tilde{\tau }}_0\right)=1\text{and}\text{SB}\end{array},$$ (8)

where $\text{Ξ}$ is a random number between 0 and 1 generated for each i. The reaction radius ${\tilde{r}}_{ij}<2.1$ is informed by previous work that demonstrates that this choice leads to the appropriate binding thermodynamics and kinetics (48). Updates to ${\omega }_{ij}\left(t\right)$ are performed randomly to avoid biasing the binding behavior based on chain index.

Inspired by the nearly symmetric geometry of a bound FIS-DNA complex (49), previous literature considers identical energy barriers of binding for each bead in the dimer, regardless of the protein’s state $\left(\Delta {\tilde{E}}_1=\Delta {\tilde{E}}_2\right)$ (39). This allowed for only one degree of freedom in the binding landscape, $\Delta {\tilde{E}}_{\text{total}}$. In this model, each of the beads in the dimer can unbind with different energy barriers, allowing for an extra degree of freedom ($\Delta {\tilde{E}}_1$ and $\Delta {\tilde{E}}_2$) and therefore a wider set of possible solutions. Recent work by Tsai et al. (50) supports this three-state model and nonsymmetric energies. $\Delta {\tilde{E}}_B=3.0$ is kept constant to ensure that the binding time, ${\tau }_B$, is equal to the diffusive time of the bead, ${\tau }_D={\tau }_B=1/{\nu }_0{e}^{\Delta {\tilde{E}}_B}=6\pi \eta a^3/\left(k_{\text{B}}T\right)$, where ${\nu }_0=1/{\tau }_0$ is the Monte Carlo update frequency.

Simulations measuring unbinding kinetics are initialized with 25 dimers that are doubly bound to the stationary DNA chain. These dimers are tagged, while competitors (other dimers or freely moving DNA chains) are initialized randomly throughout the box to obtain the desired concentration, c. When a tagged dimer diffuses a distance $6a$ from the tethered DNA strand, it becomes untagged and is removed from the system. This allows for rapid rebinding to occur, but does not increase the amount of free protein that can act as a competitor in the system. The number of tagged molecules, $n_B$, is tracked over time.

Results and Discussion

Equilibrium behavior

Facilitated dissociation dynamics are reflected in the equilibrium behavior of FIS-DNA binding, which is readily accessible in experiment via standard titration assays (51). We demonstrate this by simulation and analytical calculations of the average amount of FIS bound to DNA for a given set of energies ($\Delta {\tilde{E}}_1$ and $\Delta {\tilde{E}}_2$). Comparison of the analytical value to the simulated equilibrium behavior also verifies that the simulation reflects the intended energy landscape. The simulations are initiated with a static DNA strand of N = 50 beads and dimers randomly placed throughout the box to obtain the desired concentration c. We define dimer concentrations as the number of dimer molecules, $n_D$ per volume. After the system reaches equilibrium, the number of filled binding sites $n_p$ is averaged over >1.0 × 10⁸ time steps, which is substantially longer than the relaxation time of the system. We track the number of filled binding sites, $n_p$, as a function of $\Delta {\tilde{E}}_2$ and $\Delta {\tilde{E}}_{\text{total}}$ for various dimer concentrations c. $\Delta {\tilde{E}}_{\text{total}}=13.0$ is shown in Fig. 2 A, and a similar plot for $\Delta {\tilde{E}}_{\text{total}}=10.0$ can be seen in the Supporting Material. Symmetric energies $\left(\Delta {\tilde{E}}_1=\Delta {\tilde{E}}_2\right)$ follow the same trends as observed in Sing et al. (39).

Figure 2.

(A) Number of bound proteins $n_p$ as a function of $\Delta {\tilde{E}}_2$ for a total energy of $\Delta {\tilde{E}}_{\text{total}}=13.0$. (Solid lines) Analytical value; the data points are obtained from simulations. (B) Energy diagrams depicting the three extreme states: (left) $\Delta {\tilde{E}}_2$ is low and proteins prefer to be in the singly bound state over the doubly bound state; (center) $\Delta {\tilde{E}}_2=\Delta {\tilde{E}}_1$; (right) $\Delta {\tilde{E}}_2$ is high and proteins prefer to be in the doubly bound state over the singly bound state. To see this figure in color, go online.

The effect of nonsymmetric energies is evident in this equilibrium data. At high $\Delta {\tilde{E}}_2$ (and low $\Delta {\tilde{E}}_1$), proteins have a very low probability of moving from the doubly bound state to the singly bound state. Dimers spend most of their time bound to the DNA in the doubly bound state, taking up two binding sites along the DNA strand. The consequence is fewer proteins binding to the DNA, and the DNA being nearly saturated with doubly bound proteins at high enough concentrations. Alternatively, very low $\Delta {\tilde{E}}_2$ (and high $\Delta {\tilde{E}}_1$ in relation), the proteins have a very high probability of moving between the doubly bound state and the singly bound state (see Fig. 2 B). While proteins can be found in the doubly bound state, the time spent in that state is much less than the time spent in the singly bound state. This allows more open binding sites along the DNA strand, and at high enough concentrations, the DNA is saturated with singly bound proteins. This overbound state has been experimentally observed for a number of DNA-binding proteins (29, 52), which creates a much stiffer DNA strand.

The equilibrium adsorption behavior can be described with an analytical expression (derived in the Supporting Material) for the number of bound proteins, $\langle n_p\rangle$:

$$\langle n_p\rangle =NP\left[\frac{1+\frac{2Q+1+P}{\sqrt{{\left(1+P\right)}^2+4QP}}}{1+P+\sqrt{{\left(1+P\right)}^2+4QP}}\right],$$ (9)

where $P=e^{\Delta {\tilde{E}}_1+\tilde{\mu }}$ and $Q=e^{\Delta {\tilde{E}}_2-2}$. The value $\tilde{\mu }$ is the ideal gas chemical potential, $\tilde{\mu }={\tilde{\mu }}_0+\text{ln}\left(c\right)$, where ${\tilde{\mu }}_0=5.7$ is the constant reference chemical potential obtained from fits to the data. This is a more general version of the expression found in Sing et al. (39) for symmetric binding energies. This analytical result is plotted as lines in Fig. 2 A, and demonstrates matching with simulation for both symmetric and asymmetric binding energies.

Unbinding model

While the model proposed by Sing et al. (39) successfully described the linear concentration-dependent behavior of competing proteins in solution, later experimental work showed different behavior when the identity of the competitor was changed (34). Giuntoli et al. (34) proposed a dissociation mechanism based on DNA competitors, shown by states 0–3 in Fig. 3.

Figure 3.

Proposed unbinding mechanism. The blue dimers are proteins, the light-orange strand represents the tethered DNA strand, and the dark-orange strands represent the DNA competitors. The $k_{ij}$ values are the rate constants for moving from state i to state j. Because very little dissociation is seen without competitor in solution, there is an assumption that movement to and from state 4 is negligible, leading to states 0–3 being used to derive Eq. 10. To see this figure in color, go online.

A mean reaction time model was used to calculate the average time it takes for a protein starting in state 0 (doubly bound) to be removed from the tethered DNA strand (state 3) (53):

$$\langle {\tau }_{03}\rangle =\frac{k_{01}\left(k_{12}+k_{21}+k_{23}\right)+k_{10}\left(k_{21}+k_{23}\right)+k_{21}{k}_{23}}{k_{01}{k}_{12}{k}_{23}},$$ (10)

where $k_{ij}$ values are the rates moving from state i to state j and $k_{\text{off}}={\tau }_{03}^{-1}$. The concentration dependence is built into $k_{12}$, where the rate is multiplied by the concentration of competing DNA molecules, c. A simplified version of Eq. 10 was used to describe the unbinding kinetics observed experimentally:

$$k_{\text{off}}=c/\left(Ac+B\right),$$ (11)

where A and B are functions of the rates moving from state i to state j, $k_{ij}$; and c is the concentration of competing DNA molecules.

This result from Giuntoli et al. (34) can be fit to the observed facilitated dissociation kinetics $k_{\text{off}}$; however, the relationship to the underlying energy landscape was not explored. We connect the rate constants $k_{ij}$ to energetic and structural parameters, providing a connection between experimental observations of facilitated dissociation and molecular details of FIS-DNA interactions.

We determine the transition rates $k_{ij}$ using the values from the energy landscape in Fig. 1 B, along with system specific values, such as the binding and unbinding testing frequency, ${\nu }_0$. The rate constant $k_{01}$ is dependent only on the unbinding energy of a doubly bound protein $\left(\Delta {\tilde{E}}_{UB,2}\right)$ and ${\nu }_0$. Because both of the beads in the dimer can unbind with the same probability, there is a factor of two in the rate constant, leading to $k_{01}=2{\nu }_0{e}^{-\Delta {\tilde{E}}_{UB,2}}$. Rate constants $k_{21}$ and $k_{23}$ are similarly calculated using $\Delta {\tilde{E}}_{UB,1}$, leading to $k_{21}=k_{23}=2{\nu }_0{e}^{-\Delta {\tilde{E}}_{UB,1}}$.

The rate constant $k_{10}$ represents the transition from state 1 to state 0 (singly to doubly bound). This transition rate depends not only on the binding energy, $\Delta {\tilde{E}}_B$, but is also dependent on the number of available DNA binding positions open within $2.1a$. This leads to $k_{10}$ as a function of the expected $\Delta {\tilde{E}}_B$ and ${\nu }_0$, but indirectly as a function of $\Delta {\tilde{E}}_{UB,2}$, because this simulation value impacts how many DNA binding positions are available. A prefactor value in $k_{10}$, α, takes into account this contribution, leading to $k_{10}=\alpha {\nu }_0{e}^{-\Delta {\tilde{E}}_B}$. This prefactor decreases with increasing $\Delta {\tilde{E}}_{UB,2}$, because as $\Delta {\tilde{E}}_{UB,2}$ increases, proteins are predominantly found in the doubly bound position, leaving fewer open DNA binding sites. This results in a lower probability that a singly bound protein can bind to the DNA, leading to a smaller $k_{10}$ value.

The rate constant $k_{12}$ appears in the concentration-dependent term of $k_{\text{off}}$, $k_{12}c$. We introduce a prefactor β to the rate constant $k_{12}$ to account for the correlation of competitor binding opportunities. When there is no competitor within binding range at t, it may be more likely than average that there remains no competitor at $t+{\tau }_0$. This is a correlation effect that becomes significant at low $\Delta {\tilde{E}}_2$, where the singly and doubly bound states interchange rapidly. This leads to the expression $k_{12}=\beta {\nu }_0{e}^{\Delta {\tilde{E}}_B}$. The different competitor geometries and methods of competition lead to different binding correlations at different $\Delta {\tilde{E}}_2$.

Using the predicted $k_{ij}$ values, the full mean reaction rate equation can be simplified to the following:

$$k_{\text{off}}=\gamma \frac{\beta c}{\frac{\beta c}{2e^{-\Delta {\tilde{E}}_{1,UB}}}+\frac{2e^{-\Delta {\tilde{E}}_{2,UB}}+\alpha e^{-\Delta {\tilde{E}}_B}+e^{-\Delta {\tilde{E}}_{1,UB}}}{e^{-\Delta {\tilde{E}}_{2,UB}-\Delta {\tilde{E}}_B}}},$$ (12)

where γ is the simulation-dependent constant that takes into account the testing frequency and α and β are the aforementioned prefactors.

Facilitated dissociation simulations

We test a variety of simulation conditions for DNA competitors to verify the hypothesized model: we consider different $\Delta {\tilde{E}}_{\text{total}}$, $\Delta {\tilde{E}}_2$, and DNA competitor concentrations, defined by the number of DNA beads $\left(M\times n_s\right)$ per volume.

For each specified set of values, the number of tagged proteins in the system is tracked over time for 30 individual trajectories. In Fig. 4 A we plot the averaged number of bound proteins $\langle n_B\rangle$ for different values of $\Delta {\tilde{E}}_2$ at constant $\Delta {\tilde{E}}_{\text{total}}=13.0$ and constant $c=20.0$ ng/μL. This set of curves is an example of those seen at other values of $\Delta {\tilde{E}}_{\text{total}}$ and c, selections of which can be seen in the Supporting Material.

Figure 4.

Data from DNA facilitated dissociation simulations at $\Delta {\tilde{E}}_{\text{total}}=13.0$. (A) Unbinding curves for external $c=20\text{ng}/\mu \text{L}$. The system starts with 25 tagged proteins doubly bound to the stationary DNA strand, and DNA strands are initialized throughout the simulation box to obtain a concentration of $c=20\text{ng}/\mu \text{L}$. This corresponds to 120 competitor DNA beads. (B) When a tagged protein moves $6a$ from the tethered DNA, the protein is removed, and we observe the number of tagged proteins $\left(n_B\right)$ in the system. (Solid lines) Values calculated from Eq. 10. (Inset) Example of the $k_{\text{off}}$ for $c=20\text{ng}/\mu \text{L}$. To see this figure in color, go online.

An unbinding rate constant $k_{\text{off}}$ can be calculated from unbinding curves, like those shown in Fig. 4, via a fit to the following expression:

$$\langle n_B\rangle =n_{B,0}{e}^{-k_{\text{off}}\tilde{t}}.$$ (13)

Rate constants $k_{\text{off}}$ are plotted in Fig. 4 B as a function of concentration c and $\Delta {\tilde{E}}_2$, and at constant $\Delta {\tilde{E}}_{\text{total}}=13.0$. We account for the differences between the theoretical and simulated $k_{\text{off}}$ values by adding a $k_0$ to the theoretical $k_{\text{off}}$. The theory-based equation only accounts for the concentration-dependent dissociation pathway, and the addition of $k_0$ accounts for nonconcentration-dependent dissociation that is seen in our simulations (but not experimentally). Modified simulations that prevent unfacilitated dissociation show matching between the unmodified theory and simulated $k_{\text{off}}$ values.

The facilitated dissociation simulations were repeated with proteins as the competitor in solution, using the same method to calculate the unbinding constants $k_{\text{off}}$. Very similar behavior (both qualitatively and quantitatively) is observed and can be seen in the Supporting Material.

We can directly track the average time it takes to move from state i to state j and average it over all simulation times and trajectories to verify the theoretical $k_{ij}$ values. From these simulations, we are able to calculate the values of the prefactor α directly from the measurement of $k_{10}$, which range from 0.15 to 0.06 with increasing $\Delta {\tilde{E}}_2$. The values of β are determined from the overall $k_{\text{off}}$ behavior by fitting Eq. 12 to data, and are dependent on competitor identity. For DNA competitors, β-values range from 0.001 to 0.000045 with increasing $\Delta {\tilde{E}}_2$, and for protein competitors, β-values range from 0.00013 to 0.04 as $\Delta {\tilde{E}}_2$ increases. Both of these prefactors are independent of concentration.

When considering the unbinding constants for symmetric energies $\left(\Delta {\tilde{E}}_1=\Delta {\tilde{E}}_2\right)$ at low concentrations of either competitor, we observe the linear behavior described by previous experiments and simulations (35, 39). However, if we go to higher concentrations or nonsymmetric energies, we see that the both competitors can display the plateau off-rate seen experimentally with DNA competitors.

When at a single concentration and total energy (as in Fig. 4 A), the dissociation rate $\left(k_{\text{off}}\right)$ shows nonmonotonic behavior. It initially increases as $\Delta {\tilde{E}}_2$ increases; however, after $k_{\text{off}}$ reaches its maximum value at a turnaround $\Delta {\tilde{E}}_2$, it begins to decrease. This nonmonotonic behavior can be explained with the asymmetric unbinding energy effect included in our simulations. At low $\Delta {\tilde{E}}_2$ (high $\Delta {\tilde{E}}_1$), proteins are preferentially in the singly bound rather than the doubly bound state. The high energy barrier associated with $\Delta {\tilde{E}}_1$ must be overcome when moving from the singly bound to unbound state, leading to long unbinding times. Alternatively, at high $\Delta {\tilde{E}}_2$, the proteins are strongly bound in the doubly bound position. They must overcome the high $\Delta {\tilde{E}}_2$ barrier to become singly bound, but once they do they only need to overcome the small $\Delta {\tilde{E}}_1$ barrier to fully unbind. Both of these limits result in long unbinding times because the large energy barriers $\Delta {\tilde{E}}_1$ or $\Delta {\tilde{E}}_2$ become the bottlenecks of the dissociation mechanism.

At these two extremes, the competitors in solution do not play as much of a role in moving the tagged protein from doubly bound to singly bound to unbound. At midrange $\Delta {\tilde{E}}_2$, however, the competitors play a large role in preventing the tagged protein from moving from the singly bound state to the doubly bound state. The existence of an intermediate position is vital for facilitated dissociation, and there is no long-lived intermediate state at the two extremes of $\Delta {\tilde{E}}_2$. The fastest dissociation occurs when the competitors play the strongest role in the overall unbinding, which is where the turnaround point occurs at a midrange $\Delta {\tilde{E}}_2$ (dependent on competitor identity and $\Delta {\tilde{E}}_{\text{total}}$).

Rescaling of facilitated dissociation kinetics

All unbinding rate $k_{\text{off}}$ versus concentration c curves share similar features, such as an initial linear increase and for some conditions, an eventual plateau value (seen in the inset of Fig. 5 A). Indeed, we find that for both competitors and for all values of $\Delta {\tilde{E}}_2$, we can normalize $k_{\text{off}}\to k_{\text{off}}/k^{\ast }$ and $c\to c/c^{\ast }$ such that they fall along a single, universal curve independent of the energy landscape parameters or competitor identity (Fig. 5 A). This is also true of the theoretical fits, indicated by the solid line in Fig. 5 A. The two normalization constants, $k^{\ast }$ and $c^{\ast }$, exhibit regular trends as a function of system parameters (Fig. 5 B).

Figure 5.

(A) Universal curve for both DNA and protein competitors for $\Delta {\tilde{E}}_{\text{total}}=13.0$. Each data point is normalized by $k^{\ast }$ and $c^{\ast }$. (Solid line) Arbitrary theory curve calculated from Eq. 10. (B) The $c^{\ast }$ and $k^{\ast }$ values used to normalize the data points in (A) plotted as a function of $\Delta {\tilde{E}}_2$ on the lower x axis (for $k^{\ast }$) and $\text{ln}\left(\Delta {\tilde{E}}_{2,UB}/\xi \right)$ on the upper x axis (for $c^{\ast }$), where $\xi =\left(e^{-\Delta {\tilde{E}}_{2,UB}-\Delta {\tilde{E}}_B}\right)/\left(2e^{-\Delta {\tilde{E}}_{2,UB}}+\alpha e^{-\Delta {\tilde{E}}_B}+e^{-\Delta {\tilde{E}}_{1,UB}}\right)$. (C) Conceptual schematic describing the normalizing factors. The value $c^{\ast }$ is the concentration at which the rate from state 0 to state 2 is equal to the rate from state 2 to state 3. The value $k^{\ast }$ is the final step of dissociation, moving from state 2 to state 3. To see this figure in color, go online.

The normalization constants $k^{\ast }$ and $c^{\ast }$ can be understood in the context of Eq. 12 and two physical limiting cases. The saturated $k_{\text{off}}$ value occurs at high concentration, where the limiting step in the overall off-rate is the final dissociation of the tagged protein from the DNA strand. This can be seen mathematically with Eq. 12: when c is very large, Eq. 12 can be approximated by

$$k_{\text{off}}\approx \gamma \frac{\beta c}{\frac{\beta c}{2e^{-\Delta {\tilde{E}}_{1,UB}}}}\propto e^{-\Delta {\tilde{E}}_1}\propto e^{\Delta {\tilde{E}}_2}.$$ (14)

This relationship can be seen for both DNA and protein competitors by the red data points in Fig. 5 B. When the overall $k_{\text{off}}$ is normalized by $k^{\ast }\sim e^{\Delta {\tilde{E}}_2}$, all unbinding curves approach the same maximum off-rate.

The transition point from linearly increasing to plateau occurs at a critical concentration, $c^{\ast }$, when the rate going from state 0 to 2 (low concentration regime) is equivalent to the rate going from state 2 to 3 (high concentration regime), $k_{02}=k_{23}$, shown in Fig. 5 C. As stated above, the high concentration limit is $k_{23}\sim e^{\Delta {\tilde{E}}_2}$. The rate from state 0 to 2 can be approximated by the low concentration limit of Eq. 12:

$$k_{\text{off}}\approx \gamma \frac{\beta c{e}^{-\Delta {\tilde{E}}_{2,UB}-\Delta {\tilde{E}}_B}}{2e^{-\Delta {\tilde{E}}_{2,UB}}+\alpha e^{-\Delta {\tilde{E}}_B}+e^{-\Delta {\tilde{E}}_{1,UB}}}.$$ (15)

When the low concentration and high concentration limits of Eq. 12 are equal, we see

$$c^{\ast }\propto \frac{e^{\Delta {\tilde{E}}_2}}{\frac{\beta e^{-\Delta {\tilde{E}}_{2,UB}-\Delta {\tilde{E}}_B}}{2e^{-\Delta {\tilde{E}}_{2,UB}}+\alpha e^{-\Delta {\tilde{E}}_B}+e^{-\Delta {\tilde{E}}_{1,UB}}}}=\frac{e^{\Delta {\tilde{E}}_2}}{\xi },$$ (16)

where ξ is the low concentration limit. This relationship can be seen in the by the black data points in Fig. 5 B. Normalizing c by $c^{\ast }$ moves the transition point to the same universal location for all unbinding curves.

The different $\Delta {\tilde{E}}_2$ regimes effect the low concentration limit. At low $\Delta {\tilde{E}}_2$, $2e^{-\Delta {\tilde{E}}_{2,UB}}$ is the dominant term in the denominator due to the prefactor of 2 (and the α prefactor of the binding term), leading to $\xi \approx \beta e^{-\Delta {\tilde{E}}_B}/2$. At midrange $\Delta {\tilde{E}}_2$, the term with $\Delta {\tilde{E}}_B$ is the dominant term, even with the small α prefactor, leading to $\xi \approx \beta e^{-\Delta {\tilde{E}}_{2,UB}}$. At very high $\Delta {\tilde{E}}_2$, the $\Delta {\tilde{E}}_{1,UB}$ begins to be dominant, leading to $\xi \approx \beta e^{-\Delta {\tilde{E}}_{2,UB}-\Delta {\tilde{E}}_B}/e^{\Delta {\tilde{E}}_{1,UB}}$.

There are some differences between the simulation and experimentally observed off-rate behaviors. The energy barriers used in the simulations are much smaller than the real-world experimental values to allow us to observe dissociation on computationally accessible timescales. The nonnegligible $k_{\text{off}}$ at a concentration of 0 observed in the simulations reflects this difference. The model is otherwise not dependent on the size of the energy barriers (on the order of $k_{\text{B}}T$) and resulting timescales (<10 ms) used in the simulations. It can be extended to far longer times (>1000 s) seen experimentally (34, 35). The off-rate plateau observed in our protein-competitor simulations had not been observed in initial work (35, 39), but is consistent with more recent experiments showing saturated dissociation at sufficiently high concentrations (19).

Conclusion

Solution phase concentration-dependent dissociation has been experimentally observed for NAPs and other DNA binding proteins (19, 34, 35, 42), as well as other biological binding partners, such as DNA duplexes (36). Different concentration dependencies were observed, and multiple models have been proposed to explain the concentration dependence of different systems (34, 36, 37, 38, 39, 40, 41). This simulation work confirms the physical behavior suggested by Giuntoli et al. (34) as a result of their three-state mean reaction time model.

Our simulations allow us to understand the effect of energetic and physical parameters on the mechanism of facilitated dissociation. Regardless of the competitor identity and parameter values, all dissociation rates can be normalized to the same universal curve, which encompasses all of the different aspects of the concentration-dependent dissociation behaviors observed. At low concentrations for both simulated protein and DNA competitors, we see the linearly concentration-dependent behavior that is experimentally observed with proteins in solution (35). We also see the plateau behavior seen in DNA-competitor experiments (34), once again for both the simulated proteins and DNA with certain parameter values.

The simulations in this work were designed to mimic single-molecule experiments involving NAPs and DNA, but they can easily be expanded to other systems. By altering the binding energy $\Delta E_B$ for individual DNA beads to correspond to favorable or unfavorable binding events, we can, in principle, simulate sequence-specificity of protein binding, which is important for proteins like transcription factors (3, 54). We note that slight differences in sequence specificity can also quantitatively affect facilitated dissociation, which has previously been considered in previous work by one of the authors (39). Other changes to the system such as untethering the DNA strand, incorporating effects such as DNA allostery (55), initializing with a DNA strand that is not saturated with proteins, or adding in additional forces, all of which correspond to in vivo behaviors, can, in principle, be added. The dissociation model can also be extended to other molecules that have a multistate binding mechanism, even if molecular features are the cause of the behavior, and not the dimeric binding domains often seen in NAPs (4).

While the model used in this work is more complex than the model proposed by Sing et al. (39), we are able to capture the linearly increasing and eventual plateau of the unbinding rate trends while still retaining a robust, physical understanding of the dissociation behavior. Scaling of the low and high concentration regimes come directly from the off-rate equation (Eq. 12) that is composed of system-specific parameters. The shifting of these two different regimes with different $\Delta {\tilde{E}}_2$ can provide information about experimentally observed facilitated dissociation. Proteins that are strongly bound in the fully bound state (high $\Delta {\tilde{E}}_2/\Delta {\tilde{E}}_1$) demonstrate seemingly linear behavior, and proteins that are strongly bound in the partially bound state (low $\Delta {\tilde{E}}_2/\Delta {\tilde{E}}_1$) demonstrate rapid plateau behavior.

Our findings have shown a continuum of facilitated dissociation from concentration-dependent facilitation to saturated facilitation behavior, which has implications for NAP and nucleoid function. NAP concentrations in cells vary with different growth conditions (14), leading to different protein-DNA dissociation rates and consequently different protein-DNA association times. These varying NAP concentrations also compact the DNA differently (30, 34, 56), allowing a higher (or lower) local concentration of DNA to facilitate protein dissociation. Future experimental and simulation studies with additional NAPs can provide more information about where the dissociation rate behavior of different NAPs falls along the spectrum of facilitated dissociation.

Author Contributions

K.D. performed research and analyzed data; C.E.S. designed research; and K.D. and C.E.S. wrote the article.

Editor: Tamar Schlick.