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Figure 1

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Model illustration of a TF with the IDR and its search process.

(a) Illustration of a TF locating its target site (highlighted in orange) on the DNA strand (depicted as a green curve). Surrounding the target is a region of length $L$ , where the TF’s IDR interacts with the DNA. The TF performs 3D diffusion until it encounters and binds to this ‘antenna’ region. (b) Illustration of a TF that includes an IDR composed of AAs. On the right: a model of the TF, a polymer chain, comprising a single binding site on the DNA Binding Domain (DBD), and multiple binding sites on the IDR (also known as short linear motifs Jonas et al., 2025). (c) Once bound to the antenna, the TF performs effective 1D diffusion until it reaches its target. The 1D diffusion is via the binding and unbinding of sites along the IDR, a process we coin ‘octopusing’. $V_{1}$ is the targets’ volume, $d$ the separation, and $E_{B}$ the binding energy per binding site, where each site corresponds to a short linear motif.

Figure 2

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Design principles across various parameters: $l_{0} / d$ , $n_{b}$ , $n_{t}$ , and $E_{B}$ .

(a) Plots of $Q \equiv P_{T F} / P_{s i m p l e}$ varying with $l_{0} / d$ are shown for several combinations of $n_{t}$ and $n_{b}$ for typical values of the relevant quantities: the nucleus volume is $1 {μ m}^{3}$ , $E_{D B D} = 15$ , $E_{B} = 10$ , $V_{1} = (0.34 n m)^{3}$ , and $d = \sqrt{50} \cdot 0.34 n m$ . (b) A heatmap of $Q$ vs. $n_{b}$ and $n_{t}$ at $E_{B} = 10$ and $d = l_{0}$ . (c) Contour lines at different $E_{B}$ at $P_{T F} = 0.9$ . Blue hexagrams represent the design principle $n_{b}^{*} = n_{t}^{*}$ , with the black dashed line as a visual guide. The inset shows the dependence of $n_{b}^{*}$ on energy. (d) $Q$ for varying $E_{B}$ at $d = l_{0}$ for several $n_{b} = n_{t}$ . The vertical line represents $E_{t h}$ as determined by Equation 4, the solid curves illustrate the approximation of $Q$ obtained from Equation 3, and the horizontal line indicates where $P_{T F} = 1$ .

Figure 3

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Mean search time varying with the antenna length , $t_{3 D}$ and $t_{1 D}$ vs. at $E_{B}^{*}$ and ${\tilde{n}}^{*}$ .

The solid curves correspond to $t_{t o t a l}$ , $t_{3 D}$ and $t_{1 D}$ in Equation 5. Inset: probability density function (pdf) exponentially decays at large $t . p d f \approx e x p (- t / t_{t o t a l}) / t_{t o t a l}$ (black solid line). The unit $t_{0}$ represents the time for one AA to diffuse $1 b p$ . $a = 0.5 b p$ , $d = 10 b p$ , and $l_{0} = 5 b p$ .

Figure 4

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Comparisons with experimental results.

(a) Relative binding probability varies with the truncation length of the IDR, which quantitatively agrees with the experimental data in Brodsky et al., 2020, where we only vary $E_{D B D}$ (binding energy of DBD) to ensure that the maximal value reaches 1 at zero truncation length. Antenna length $L = 1000 n m$ , $E_{D B D} = 22$ and $E_{B} = 11$ . Other parameters are set the same as in Figure 2. (b) The search time estimated by our model (from Equation 5 and divided by a TF copy number of 100) is quantitatively comparable with the experimental results (Larson et al., 2011), as guided by the shaded area. c, The on-rate $k_{o n} = t_{t o t a l} / V_{c}$ and off-rate $k_{o f f} = (1 - P_{T F}) / (P_{T F} t_{t o t a l})$ varying with the IDR length, indicating that $k_{D} \equiv k_{o f f} / k_{o n}$ ranges from $0.01$ to 10 nM, with its variation primarily dominated by the off-rate.

© 2020, Elsevier. Panel A was reprinted with permission from Brodsky et al., 2020 with permission from Elsevier. It is not covered by the CC-BY 4.0 license and further reproduction of this panel would need permission from the copyright holder.

Appendix 1—figure 1

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$f (r, l_{0})$ aligns well with Equation 10 (solid curves) for moderate $m$ at various.

$b_{0}$ Inset: $a_{e f f}^{2} / a_{0}^{2}$ vs. $b_{0} / a_{0}$ from Equation 12. (b) Equation 10 serves as a good approximation (solid lines) even at small $m$ .

Appendix 2—figure 1

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A diagram of the bound configurations at $n = 3$ .

Appendix 3—figure 1

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Maximal $Q$ is observed at $l_{0} \sim d$ for all combinations of $n_{b}$ and $n_{t}$ .

Appendix 4—figure 1

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Binding probability varying with $n_{b}$ and $n_{1}$ .

(a) $P_{T F}$ varying with $n_{b}$ at $n_{t} = 12$ . $P_{T F}$ initially increases rapidly, until eventually saturating. (b) $P_{T F}$ varies non-monotonically with $n_{t}$ . $P_{T F}$ slightly decreases with increasing $n_{t}$ at $n_{t} > n_{b}$ . Three examples at $n_{b} = 7, 8, 9$ are displayed. $P_{T F}$ is denoted by pentagrams at $n_{t} = n_{b}$ .

Appendix 5—figure 1

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Contour lines at $P_{T F} = 0.9$ and $P_{T F} = 0.5$ on the plane defined by the number of binding sites, $n_{b}$ , and the number of binding targets, $n_{t}$ .

The black dashed line is added to guide the relationship $n_{b} = n_{t}$ .

Appendix 6—figure 1

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Plots of the affinity enhancement, quantified by the ratio $Q \equiv P_{T F} / P_{s i m p l e}$ , for the case where IDR sites as well as their targets are randomly positioned on the IDR and DNA, respectively.

See the analogous Figure 2 in the main text for the case of homogeneously distributed IDR sites and targets. (a) $Q$ is maximal when $l_{0} \sim d$ (shaded region) for different values of $n_{t} = n_{b}$ . The average distance between IDR sites is $l_{0}$ , and the average distance between IDR targets is $d$ . (b) A heatmap of $Q$ at $E_{B} = 10$ and $l_{0} = d$ . (c) Contour lines at different $E_{B}$ at $P_{T F} = 0.5$ exhibit an approximate symmetrical L-shape, which indicates $n_{b}^{*} = n_{t}^{*}$ , as shown by the red triangles. $n_{b}^{*}$ decreases with increasing $E_{B}$ . (d) $Q$ varies with $E_{B}$ for several different constants of $n_{t} = n_{b}$ . The characteristic energy $E_{t h}$ , as predicted by Equation 4 in the main text, corresponds to the region where a notable increase is observed.

Appendix 7—figure 1

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Demonstration of a single round of 3D-1D search in simulations.

(a) A trajectory (red) of TF in the cell nucleus (light blue). is the average distance of the TF to the antenna, and $X_{c}$ is the position of the center of mass along the antenna direction. The trajectory is displayed with a time interval of $10^{6} t_{0}$ . (b) The average number of rounds the TF hits the antenna, $n_{h i t s}$ , before reaching the DBD target. The optimal mean total search time is obtained at $L^{*} = 300 b p$ .

Appendix 7—figure 2

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Search times as a function of $E_{B}$ and $\tilde{n}$ .

(a) $t_{t o t a l}$ , $t_{3 D}$ , and $t_{1 D}$ vs. at fixed $L^{*}$ and ${\tilde{n}}^{*}$ . The red solid line is the prediction of $t_{3 D}$ from Equation 5, and the blue line (also in the inset) is the exponential fit for $t_{1 D}$ . The black curve in the inset is the number of rounds the TF hits the antenna, $n_{h i t s}$ , vs. $E_{B}$ . (b) $t_{t o t a l}$ , $t_{3 D}$ and $t_{1 D}$ vs. $\tilde{n}$ at fixed $L^{*}$ and $E_{B}^{*}$ . The red and blue solid lines represent the predictions of $t_{3 D}$ and $t_{1 D}$ from Equation 5, respectively. $n_{h i t s}$ varying with $\tilde{n}$ is shown in the inset.

Appendix 9—figure 1

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Left: an example of the target configuration (the antenna) generated by the worm-like chain model, where $L = 300 b p$ , $d = 10 b p$ , and the persistence length also equals $10 b p$ .

Right: Mean total search times weakly vary with persistence length when targets are generated using the worm-like chain model. The values are represented by the black dots, with parameters set to $E_{B} = 11$ , $L = 300$ bp, and $\tilde{n} = 4$ . For each persistence length, we obtained ten mean total search times by considering ten different target configurations. The mean total search times do not exhibit significant differences compared to that for targets arranged linearly (indicated by the horizontal orange line).

Appendix 10—figure 1

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Search in a complex DNA configuration.

(a) Configuration of DNA (in red) at $ν = 0.008$ with a complex structure and a trajectory of TF (in black). A TF acts as a point searcher that diffuses in 3D space and becomes trapped once it hits the DNA, which is $1 n m$ wide. It detaches at a rate of $\frac{D_{3}}{1 n m^{2}} \exp (- E_{n o n s p} / k_{B} T)$ . (b) MSD at different $E_{n o n s p}$ .

Appendix 11—figure 1

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Property of 1D octopusing walk.

(a) $W_{n, n + 1}$ (or $W_{n, n - 1}$ ) denoted by ‘on’ (or ‘off’) varies with $n$ for $E_{B} = 6$ and $\tilde{n} = 12$ . Inset: Mean square displacement (MSD) suggests diffusive motion, as indicated by the dashed line following $M S D \propto t$ . (b) Steady probability distribution of bound sites.

Videos

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posterframe for video — Video of the 1D octopusing walk simulation.

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Wencheng Ji
Ori Hachmo
Naama Barkai
Ariel Amir

(2025)

Design principles of transcription factors with intrinsically disordered regions

eLife 14:RP104956.

https://doi.org/10.7554/eLife.104956.3

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