Figure 1

(Color online) Discrete wormlike chain model for proteins binding to DNA.Reuse & Permissions

Figure 2

(Color online) Force-extension curves of DNA-protein complex for different chemical potentials $\mu$ at different cooperativity parameters $\eta$. (a)–(d) show results for $\eta =0$, 2, 3, and 4, respectively. Here, the bending stiffness of DNA is $a=10$ and that of the DNA-protein complex is $a^{\prime }=50$. In each panel, the chemical potentials are, for curves from left to right, $\mu =-\infty$ (naked DNA, leftmost black curve), $-5.0$ (red curve), $-4.0$ (green curve), $-3.0$ (blue curve), $-2.0$ (pink curve), 0.0 (orange curve), $+\infty$ (saturated DNA, rightmost black curve). Note that in (d), the strong cooperativity $\left(\eta =4\right)$ makes the curve with high protein concentration $\left(\mu =0.0\right)$ approach the protein-saturated $\left(\mu =+\infty \right)$ DNA curve in the force range shown (0–10 pN).Reuse & Permissions

Figure 3

(Color online) Fluctuation of extension versus force, for DNA-protein complex as a function of chemical potential $\mu$ and cooperativity $\eta$. (a)–(d) show results for $\eta =0$, 2, 3, and 4, respectively; all parameters are the same as in Fig. 2. Color-coded values of $\mu$ are shown in each panel adjacent to corresponding curves. (a) For zero cooperativity, the fluctuations are quenched by force and are also reduced by increasing protein concentration (increasing $\mu$). (b)–(d) For nonzero cooperativity, peaks appear in the extension fluctuation corresponding to rapid increases in extension with force observed in Fig. 2. Note that in (d) the curve for $\mu =0.0$ is coincident with the saturated $\mu =+\infty$ curve.Reuse & Permissions

Figure 4

(Color online) Protein occupation curves for different chemical potentials $\mu$ at different cooperativity parameters $\eta$, as in Figs. 2, 3. In each panel, the chemical potentials are, from bottom curve to top curve, $\mu =-\infty$ (naked DNA, bottommost curve), $-5.0$, $-4.0$, $-3.0$, $-2.0$, 0.0, and $+\infty$ (saturated DNA, topmost curve), with the same color coding as in Figs. 2, 3. Increasing force drives down binding of DNA-bending proteins. As cooperativity $\eta$ is increased, the transition between protein-bound DNA and naked DNA becomes sharper. Note that in (d), the curve with chemical potential $\mu =0.0$ superposes on the protein-saturated curve $\left(\mu =+\infty \right)$.Reuse & Permissions

Figure 5

(Color online) Protein occupation fluctuation as a function of force for different chemical potentials $\mu$ (color codes and $\mu$ values are shown in each panel) and for different cooperativity parameters [(a)–(d) correspond to $\eta =0$, 2, 3, and 4]. In each panel, the bottommost black curve has chemical potential $\mu =-\infty$ (naked DNA) and $+\infty$ (saturated DNA). In (a) and (b), the list of $\mu$ values applies to the upper curves in the order indicated; otherwise, the labels apply to the curve or peak to which they are adjacent. In (d), the curve with chemical potential $\mu =0.0$ superposes onto the protein-saturated case $\left(\mu =+\infty \right)$ over the force range shown.Reuse & Permissions

Figure 6

(Color online) (a) Extension, (b) extension fluctuation, (c) protein occupation, and (d) protein occupation fluctuation versus chemical potential for different cooperativities $\eta$ (different curves) at a fixed applied force (0.5 pN). Cooperativities for curves with transitions or peaks in right-to-left order are $\eta =0$ (rightmost, black curve), 1 (red curve), 2 (green curve), 3 (blue curve), 4 (pink curve), and 5 (leftmost, orange curve). As cooperativity is increased, the transitions in extension and bound protein number shift to lower chemical potential, and corresponding fluctuations develop larger and sharper peaks.Reuse & Permissions

Figure 7

(Color online) Normalized extension $R_N$ vs DNA-binding protein number per binding site $N_P/N$ for different cooperativities $\eta$. Different panels show a series of fixed forces of (a) 0.01, (b) 0.2, (c) 0.5, and (d) 5.0 pN. In each panel curves show results for different cooperativities; from top to bottom curves correspond to $\eta =0$ (topmost, black curve), 1 (red curve), 2 (green curve), 3 (blue curve), 4 (pink curve), and 5 (bottommost, orange curve). For the lowest force (0.01 pN) the normalized extension does not report well on the protein occupation, but for the larger forces the $R_N$ curves converge toward the straight line $R_N=N_P/N$.Reuse & Permissions

Figure 8

(Color online) Peak value of slope of normalized extension $R_N$ versus chemical potential as a function of cooperativity $\eta$. (a)–(d) indicate different fixed forces of 0.01, 0.2, 0.5, and 5.0 pN, respectively. Different curves show different values of $\eta$: from rightmost lowest peak to leftmost highest peak $\eta =0$ (rightmost, black curve), 1 (red curve), 2 (green curve), 3 (blue curve), 4 (pink curve), and 5 (leftmost, orange curve). Note that peaks are nearly the same for 0.2 and 0.5 pN, and only slightly different for 0.01 and 5.0 pN.Reuse & Permissions

Figure 9

(Color online) Estimated cooperativity parameter ${\eta }_{\text{eff}}$ and peak position ${\mu }_{\text{peak}}$ extracted from analysis of peaks in $R_N\left(\mu \right)$ curves. (a) Estimated cooperativity parameter ${\eta }_{\text{eff}}$ versus microscopic $\eta$ value for a series of fixed forces shown by different thin colored curves; from bottom to top curves are for 0.01 (red curve), 0.2 (green curve), 0.5 (blue curve), and 5.0 (pink curve) pN. Over a broad force range ${\eta }_{\text{eff}}$ is an accurate reporter on the actual microscopic $\eta$ parameter; note that the results for 0.2 and 0.5 pN are very close to ${\eta }_{\text{eff}}=\eta$ (heavy black line). (b) Peak position ${\mu }_{\text{peak}}$ versus $\eta$ for the same series of fixed forces as in (a). The peak position is appreciably shifted away from ${\mu }_{\text{peak}}=-\eta$ (heavy black line) due to force-extension free-energy shift due to DNA bending.Reuse & Permissions

Figure 10

(Color online) A DNA polymer configuration with exactly two proteins present at fixed positions from one another, $n=11$ segments apart.Reuse & Permissions

Figure 11

(Color online) Force-generated cooperativity. (a) Extension change versus interprotein distance for a series of fixed forces (different curves). Extension change is shown relative to the limit where the two proteins are far apart on the DNA (see text). At the left, the top curve has force $f=0.1\text{pN}$ (green curve), then in decreasing order the curves show forces $f=0.04$ (red curve), 0.5 (blue curve), 1.0 (pink curve), 5.0 (orange curve), and 0.002 (black curve) pN. Note that these curves are nonmonotonic and can change sign with protein spacing, reflecting that either extension increase or decrease can occur as the proteins are moved closer together. Inset shows extension changes versus applied force (again relative to distant-protein limit), for proteins on adjacent binding sites. The extension change shows a nonmonotonic dependence on force but is always positive, corresponding to an increase in total complex length relative to the case where the two proteins are far apart. (b) Force-generated cooperative interaction potential extracted from free energy versus interprotein distance for different forces. At the left, top curve has force $f=0.002\text{pN}$ (black curve), then in the decreasing order the curves have forces $f=0.04$ (red curve), 0.1 (green curve), 0.5 (blue curve), 1.0 (pink curve), and 5.0 (orange curve) pN. Inset shows $\text{ln}\left(-\Delta \beta F\right)$ versus interprotein distance at $f=1.0\text{pN}$ (red crosses) and a linear fit (blue line), showing that the interaction is close to exponential. (c) Force-generated cooperative interaction correlation length versus applied force (black plus signs) and a power-law fit to the high-force behavior (red line) showing that correlation length varies $\approx {\left(k_{B}TA/4f\right)}^{1/2}$. (d) Force-generated cooperative interaction amplitude versus applied force: the long-range exponential decay interaction amplitude $U\left(f\right)$ (top black plus signs), the short-range interaction well depth $V_0\left(f\right)$ (bottom red crosses), and a power-law fit to them (blue lines). Neither of the two amplitudes is well described by a simple power law and $U\left(f\right)$ increases more rapidly than $f^{1/2}$ for large forces.Reuse & Permissions