Design principles of transcription factors with intrinsically disordered regions

The experiment described in Brodsky et al., 2020 studied the binding affinity of wild-type and mutated TFs to their corresponding targets. They found that shortening the IDR length results in weaker affinity. To qualitatively model the binding affinity of a TF to its target, we utilize a well-known physical model of polymers, the Rouse Model (Prince, 1953; Doi and Edwards, 1988; used in the subsequent dynamical section as well). See Materials and methods. We note that our conclusions do not hinge on this choice, and our results hold also when using other polymer models (see Appendix Appendix 1). Broad applicability of probability density $f(r,l)$. Since the IDR is thought to recognize a relatively short flanking DNA sequence of the DBD target (Brodsky et al., 2020), we model a DNA segment as a straight-line ‘antenna’ region (Shimamoto, 1999; Hu and Shklovskii, 2007; Mirny et al., 2009; Castellanos et al., 2020), where the IDR targets reside. We assume that the interaction between TF and DNA outside the antenna region is negligible. Note that the interaction between IDR and DNA is a specific interaction, with genomic localization determined by multivalent interactions mediated by hydrophobic residues (Jonas et al., 2023; Mindel et al., 2024).

To gain an intuition into the binding probability of TF, we first study a TF with a single IDR site. From here on, energies will be measured in units of $k_{B}T$, to simplify our notation. We assume that the IDR has only one corresponding target. There are four possible states: neither the DBD nor the IDR is bound, the IDR is bound, the DBD is bound, and both are bound, which correspond to $P_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$, $P_{\mathrm{I}\mathrm{D}\mathrm{R}}$, $P_{\mathrm{D}\mathrm{B}\mathrm{D}}$, and $P_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}}$, respectively. The probability that the TF is bound at the DBD target is the sum of $P_{\mathrm{D}\mathrm{B}\mathrm{D}}$ and $P_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}}$. In thermal equilibrium, each of the 4 terms is given by the Boltzmann distribution, which can be solved analytically by integrating out the additional degrees of freedom associated with the polymer (Appendix 2. Derivation of the binding probability). To better understand the associated design principles, it will be helpful to compare this probability with that of a ‘simple’ TF (i.e. without the IDR), which we denote by $P_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}$. We find that:

where $\mathcal{P}$ is an enhancement factor, governing the degree to which the IDR improved the binding affinity. Importantly, we find that:

where $f(d,l_0)={\left(\frac{3}{2\pi l_0^2}\right)}^{3/2}\exp \left(-\frac{3d^2}{2l_0^2}\right)$, is the probability density, with units of inverse volume, $V_1$ is the targets’ volume, on the order of ${1-10\mathrm{b}\mathrm{p}}^3$, as shown in Figure 1c, $-E_B$ is the binding energy for per IDR target ($E_B$ is always positive), $d$ is the distance between the DBD target and the IDR target, and $l_0$ is the distance between the DBD and the IDR binding site on the TF ($l_0$ and $d$ are also the distances for neighboring IDR targets and neighboring IDR sites, respectively, in the case of multiple IDR targets and sites).

This shows that the presence of the IDR always increases the binding affinity since $\mathcal{P}$ is always positive. However, the magnitude of enhancement can be negligible: For instance, even if $E_B$ is large, the term $f(d,l_0)$, associated with the entropic penalty (because the end-to-end distance is fixed, reducing the degrees of freedom compared to a free polymer), could lead to small values of $\mathcal{P}$.

The exponential decay of $\mathcal{P}$ with $d^2$ (through the term $f(d,l_0)$), indicates that placing the IDR targets close to the DBD target is preferable. However, due to physical limitations, $d$ cannot be arbitrarily small and in reality will have a lower bound. Once $d$ is fixed, $\mathcal{P}$ is maximal at $l_0=d$, as can be gleaned from Equation 2. This condition minimizes the free energy penalty associated with the entropic contribution of the term $f(d,l_0)$.

As we shall see, this design principle is also intact when multiple IDR sites and targets are involved, which is the scenario we discuss next.

A priori, it is unclear whether having more binding sites ($n_b>1$) on the IDR or more targets ($n_t>1$) on the DNA will be beneficial or not. We therefore sought to investigate how the binding affinity depends on these parameters, as well as the binding energy and positions of the targets and binding sites. This generalization requires taking into account all the possible number of bindings, $n$ (ranging from 0 to $min(n_b,n_t)$), and all possibilities for binding orientations for each $n$, $(\genfrac{}{}{0ex}{}{n_b}{n})(\genfrac{}{}{0ex}{}{n_t}{n})n!$ distinct configurations in total. For each of them, we may analytically integrate out the polymer degrees of freedom and calculate explicitly the contribution to the partition function (Appendix 2. Derivation of the binding probability). Summing these terms together, numerically, allows us to explore how the different parameters affect the binding specificity, which we describe next. To characterize the binding affinity, we resort again to the ratio $Q\equiv P_{\mathrm{T}\mathrm{F}}/P_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}$, measuring the effectiveness of the binding in comparison to the case with no IDRs. The length of TFs in our considerations, which varies with $n_b$, is comparable to that in experiments (Brodsky et al., 2020).

We find the design principle: $l_0=d$, identical to what we showed for the case of a single IDR binding site. Specifically, $Q$ is maximal at $l_0\sim d$ for all combinations of $n_b$ and $n_t$. Figure 2a displays this principle for several combinations of $n_b$ and $n_t$. All combinations are provided in Appendix 3—figure 1. This is plausible since for the IDR to be bound to the DNA at multiple places without paying a large entropic penalty, the distance between target sites should be compatible with the distance between binding sites.

Figure 2

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We therefore set $l_0=d$ and search for other design principles. Figure 2b shows a heatmap of $Q$ varying with $n_b$ and $n_t$. At large $n_t$, as $n_b$ increases, the value of $P_{\mathrm{T}\mathrm{F}}$ exhibits an initial substantial increase, followed by a gradual rise until it reaches saturation (shown explicitly in Appendix 4—figure 1).

A similar behavior is also observed when varying the value of $n_t$. (When $n_t$ exceeds $n_b$, the contribution from the states where the IDR is bound but the DBD is unbound increases with increasing $n_t$, which slightly lowers $P_{\mathrm{T}\mathrm{F}}$; see Appendix 4—figure 1.)

In fact, the heatmap shown in Figure 2b shows that the value of $P_{\mathrm{T}\mathrm{F}}$ is governed, approximately, by the minimum of $n_b$ and $n_t$ (this is reflected in the nearly symmetrical ‘L-shapes’ of the contours). This places strong constraints on the optimization and suggests that there will be little advantage in increasing the value of either $n_b$ or $n_t$ beyond the point $n_b=n_t$, revealing another design principle (under the assumption that it is preferable to have smaller values of $n_b$ and $n_t$ for a given level of affinity, in order to minimize the cellular costs involved).

We verify this principle holds at additional values of $E_B$. Figure 2c shows that the contours corresponding to $P_{\mathrm{T}\mathrm{F}}=0.9$ at varying values of $E_B$ all take an approximate symmetrical L-shape, thus supporting the design principle $n_b=n_t$. For a given $E_B$, we denote the optimal values as $n_b^{\ast }=n_t^{\ast }$, and show them as hexagrams in the plot.

We further find that $n_b^{\ast }$ decreases with increasing $E_B$, as displayed in the inset. These relationships also hold at $P_{\mathrm{T}\mathrm{F}}=0.5$ (Appendix 5. Robustness of the relationships: $n_b^{\ast }\sim n_t^{\ast }$ and $n_b^{\ast }$ decreases with increasing $E_B$).

We examine the variation of $Q$ with respect to $E_B$ while keeping $n_b=n_t$ constant. In Figure 2d, as the binding energy $E_B$ increases, $Q$ increases from a value of $\sim 1$ to its saturation value, $1/P_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}$, in a switch-like manner: that is, there is a characteristic energy scale, $E_{\mathrm{t}\mathrm{h}}$. To estimate $E_{\mathrm{t}\mathrm{h}}$, when $Q\ll 1/P_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}$ (i.e. $P_{\mathrm{T}\mathrm{F}}\ll 1$), we approximate the expression of $Q$ as:

where $l_{\mathrm{o}\mathrm{r}\mathrm{n},i}$ and $r_{\mathrm{o}\mathrm{r}\mathrm{n},i}$ denote the segment length between two adjacent bound sites and the distance associated with their respective targets, respectively, and ‘orn’ represents the orientation of the given configuration (Appendix 2. Derivation of the binding probability). Equation 3 reduces to Equation 1 when $n_t=n_b=1$. To substantially increase $Q$, the configuration that contributes the most to $Q$, would satisfy $(\mathcal{P}(E_B,d,d){)}^{n_b}\gg 1$. Thus, the characteristic energy $E_{\mathrm{t}\mathrm{h}}$ could be estimated using the formula $\mathcal{P}(E_{\mathrm{t}\mathrm{h}},d,d)=1$, resulting in:

where $\varphi =V_1^{1/3}/d$ represents the proportion of IDR targets within the antenna. Equation 4 is a direct consequence of the energy-entropy trade-off. Increasing $\varphi$ leads to a decrease in entropy loss due to binding the IDR, subsequently reducing the demands on binding energy. Given that ${\varphi }^2=0.02$, $E_{\mathrm{t}\mathrm{h}}$ is about $8.5k_{B}T$. The vertical dashed line in Figure 2d, obtained from Equation 4, effectively captures the region of significant increase for large values of $n_b$. Hence, we obtain the third design principle: $E_B>E_{\mathrm{t}\mathrm{h}}$, which ensures the significant binding advantage of TFs with IDRs.

So far, we have found that the binding design principles for TFs with IDRs are as follows: (i) target distance $d$ should be as small as possible, (ii) IDR segment binding distance $l_0$ should be comparable to $d$, (iii) it is preferable to have fewer binding sites $n_b^{\ast }$ and binding targets $n_t^{\ast }$ as the binding strength $E_B$ increases, with $n_b^{\ast }\sim n_t^{\ast }$, and (iv) should be larger than a threshold that solely depends on the proportion $\varphi$ of IDR targets within the antenna, as indicated by Equation 4. In our estimation, with ${\varphi }^2=0.02$, we obtain $E_{\mathrm{t}\mathrm{h}}\approx 8.5k_{B}T$. The above principles have also been validated in scenarios involving Poisson statistics of binding site distances and target distances, thus indicating their robust applicability (Appendix 6—figure 1).

In this section, we study the design principles of the TF search time, a dynamic process. Here, we confine the TF with $\tilde{n}\equiv n_b+1$ sites ($n_b$ IDR sites and one DBD site) to a sphere of radius $R$, the cell nucleus, and use over-damped coarse-grained molecular dynamics (MD) simulations. The sites and targets interact via a short-range interaction. The DBD target with a radius $a$ is placed at the sphere’s center. See the sketch in Figure 1a. Once the DBD finds its target, the search ends. We denote the mean total search time (i.e. the mean first passage time, MFPT), as $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$. We estimate it by averaging 500 simulations, each starting from an equilibrium configuration of the TF at a random position in the nucleus. Simulation details are provided in the Materials and Methods section.

By conducting comprehensive simulations of $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ across various parameter values of $L$, $E_B$, and $\tilde{n}$, we were able to identify its minimum value, which is approximately at $L^{\ast }=300\mathrm{b}\mathrm{p}$, $E_B^{\ast }=11$, and ${\tilde{n}}^{\ast }=4$ for $R=500\mathrm{b}\mathrm{p}\approx 1/6\mu \mathrm{m}$. Here, one base pair ($\mathrm{b}\mathrm{p}$) is equal to $0.34\mathrm{n}\mathrm{m}$. To speed up the simulations, we utilized several computational tricks, including adaptive time steps and treating the TF as a point particle when it is far from the antenna (see Materials and methods). Despite this, the simulations require 10⁵ CPU hours due to the IDR trapped in the binding targets on the antenna, which results in the slow dynamics as found in the previous simulation works (Vuzman and Levy, 2010; Marcovitz and Levy, 2013). The probability distribution function (pdf) of individual search times exponentially decays with a typical time equal to $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, see inset of Figure 3. This indicates that $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ provides a good characterization of the search time.

Figure 3

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We find that $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ varies non-monotonically with $L$ at fixed $E_B^{\ast }$ and ${\tilde{n}}^{\ast }$ as shown in Figure 3. Once a TF attaches to one of its targets, it will most likely diffuse along the antenna until reaching the DBD target (Appendix 7. TF walking along the antenna once it attaches, and $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ vs. $E_B$ and vs. $\tilde{n}$). It means that the optimal search process is a single round of 3D diffusion followed by 1D diffusion along the antenna. In Appendix 8, a single round of 3D-1D search is typically the case in the optimal search process., by analytically calculating the mean total search time using a coarse-grained model, we find that a single round of 3D-1D search is typically the case in the optimal search process. Note that our model differs profoundly from the conventional facilitated diffusion (FD) model in bacteria: In the latter, the TF is assumed to bind non-specifically to the DNA. Within our model, the antenna length $L$ can be tuned by changing the placement of the IDR targets on the DNA, making it an adjustable parameter rather than the entire genome length. As opposed to the FD model, which requires numerous rounds of DNA binding-unbinding to achieve minimal search time, within our model, the minimal search time is achieved for a single round of 3D and 1D search.

Interestingly, when we observe the motion of a TF along the antenna, IDR sites behave like ‘tentacles’, as illustrated in Figure 1c, constantly binding and unbinding from the DNA. We refer to this 1D walk as ‘octopusing’, inspired by the famous ‘reptation’ motion in entangled polymers suggested by de Gennes (de Gennes and Leger, 1982; see Video 1). We note that previous numerical studies Vuzman and Levy, 2010; Vuzman and Levy, 2012 used comprehensive, coarse-grained, MD simulations to demonstrate that including heterogeneously distributed binding sites on the IDR could facilitate intersegmental transfer between different regions of the DNA, which is reminiscent of our octopusing scenario.

Video 1

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The mean 3D search time, $t_{3\mathrm{D}}$, and the mean 1D search time, $t_{1\mathrm{D}}$ are also shown in Figure 3. To evaluate $t_{3\mathrm{D}}$, we take advantage of several insights: first, if the TF gyration radius, $r_p=\sqrt{\tilde{n}/6}{l}_0$, is comparable or larger than the distance between targets, $d$, we may consider the chain of targets as, effectively, a continuous antenna. Making the targets denser would hardly accelerate the search time, reminiscent of the well-known chemoreception problem where high efficiency is achieved even with low receptor coverage (Berg and Purcell, 1977). Second, when the distance between the TF and the antenna is smaller than $r_p$, one of the binding sites on the TF would almost surely attach to the targets. This allows us to approximately map the problem to the MFPT of a particle hitting an extremely thin ellipsoid with long axis of length $L$ and radius $r_p$, the solution of which is $\ln (L/r_p)/L$ (Berg, 2018). We find that this scaling captures well the dependence of $t_{3\mathrm{D}}$ on the parameters, with a prefactor $\alpha \sim O(1)$.

For the 1D search from a single particle picture, since the TF is unlikely to detach once it attaches, it performs a 1D random walk. So, taken together, the mean total search time is

where $D$ is the 3D diffusion coefficient of one site. For a TF with $\tilde{n}$ sites, its 3D diffusion coefficient, $D_3$, is $D/\tilde{n}$ and $\beta$ are fitting parameters.

The semi-analytical formulas of $t_{1\mathrm{D}}$, $t_{3\mathrm{D}}$, and $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, denoted by the solid lines in Figure 3, agree well with the simulation results for $\alpha =0.5$ and $\beta =14.5$.

The theoretical result for the time spent in 3D was originally calculated for a point searcher (Berg, 2018). In that case, a value of $\alpha =2/3$ was obtained. $\beta$ is larger at larger $E_B$ (see Appendix 7. TF walking along the antenna once it attaches, and $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ vs. $E_B$ and vs. $\tilde{n}$).

We also examine the dependence of $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ on the key parameters $E_B$ and $\tilde{n}$, in the vicinity of the optimal parameters which minimize the total search time (Appendix 7. TF walking along the antenna once it attaches, and $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ vs. $E_B$ and vs. $\tilde{n}$). We find that the affinity of TFs to the antenna decreases rapidly with decreasing the IDR length (Brodsky et al., 2020) and $E_B$. Consequently, when $\tilde{n}<{\tilde{n}}^{\ast }$ or $E_B<E_B^{\ast }$, multiple search rounds are needed to reach the DBD target, resulting in a significant escalation of $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$. When $\tilde{n}>{\tilde{n}}^{\ast }$, $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, $t_{3\mathrm{D}}$, and $t_{1\mathrm{D}}$ agree with Equation 5. When $E_B>E_B^{\ast }$, since $t_{3\mathrm{D}}$ is independent of $E_B$, the variation in $t_{\mathrm{t}\mathrm{o}\mathrm{S}7\mathrm{t}\mathrm{a}\mathrm{l}}$ stems from $t_{1\mathrm{D}}$ which exponentially increases with $E_B$.

The minimal search time for a simple TF is $t_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=R^3/(3Da)$ (Berg, 2018).

For the TF with an IDR, by minimizing the expression in Equation 5, we find that the optimal length approximately scales as $L^{\ast }\propto R$, and therefore the minimal search time, $t_{min}$, is approximately proportional to $R^2$.

At $R=500\mathrm{b}\mathrm{p}\approx 1/6\mu \mathrm{m}$, the search time is more than ten times shorter than a simple TF, despite the latter’s diffusion coefficient being several times larger. Based on our analytic result, we expect $\sim 50$ fold enhancement for the experimental parameters of yeast note that we assume the scaling $t_{min}\sim R^2$ still holds in this regime; this formula is approximate since it does not take into account the persistence length of DNA (Guilbaud et al., 2019), which is several-fold shorter than the optimal antenna length $L^{\ast }$. We have verified that reducing the DNA persistence length, which promotes increased DNA coiling, results in only a modest increase in mean search time. Even under extreme coiling conditions, the increase remains below 30% of the baseline value, as detailed in Appendix 9. Robustness of mean total search time with changing the antenna geometry. Converting $t_{min}$ into a second-order on-rate yields approximately ${10}^8-{10}^9{\mathrm{M}}^{-1}{s}^{-1}$, assuming $D\approx {10}^{-13}{m}^2/s$ and a TF copy number of 100. We also verified that the 3D diffusion coefficient slightly changes under a complex DNA configuration, where a point-searcher TF non-specifically interacts with the entire DNA, indicating the robustness of our optimal search process (see Appendix 10. MSD in a complex DNA configuration).

Next, we characterize the dynamics of the octopusing walk (Appendix 11. Properties of the octopusing walk). During this process, different IDR binding sites constantly bind and unbind from the antenna, leading to an effective 1D diffusion. A priori, one may think that the TF faces conflicting demands: it has to rapidly diffuse along the DNA while also maintaining strong enough binding to prevent detaching from the DNA too soon. Indeed, within the context of TF search in bacteria, this is known as the speed-stability paradox (Slutsky and Mirny, 2004). Importantly, the simple picture of the octopusing walk explains an elegant mechanism to bypass this paradox, achieving a high diffusion coefficient with a low rate of detaching from the DNA. This comes about due to the compensatory nature of the dynamics: there is, in effect, a feedback mechanism whereby if only a few sites are bound to the DNA, the on-rate increases, and similarly, if too many are bound (thus prohibiting the 1D diffusion) the off-rate increases. Remarkably, this mechanism remains highly effective even when the typical number of binding sites is as low as two. Further details are provided in Appendix 11. Properties of the octopusing walk.

In Brodsky et al., 2020, the binding probability of the TF Msn2 in yeast was measured, where the native IDR was shortened by truncating an increasingly large portion of it. The results indicate that binding probability decreases with increasing truncation of the IDR, as illustrated in Figure 4a. We utilized our model to calculate the relative binding probability compared to $P_{\mathrm{T}\mathrm{F}}$ at a zero truncating length. Using the DBD binding energy $E_{\mathrm{D}\mathrm{B}\mathrm{D}}$ as our only fitting parameter, our model achieves quantitative agreement with the experimental data. In Figure 4b, we estimate the search time as a function of $L$ for $R=1\mu m$ and $D_0={10}^{-10}{m}^2/s$ for a single AA. The estimated timescales are comparable to empirical results, which are 1 to 10 seconds (Kumar et al., 2023; Larson et al., 2011), as indicated by the shaded area. We have also estimated the on-rate $k_{\mathrm{o}\mathrm{n}}$ and off-rate $k_{\mathrm{o}\mathrm{f}\mathrm{f}}$ of the TF as they vary with IDR length, as shown in Figure 4c. In the kinetic proofreading mechanism (Hopfield, 1974), specificity is associated with the off-rate, but recent studies (Marklund et al., 2022) suggest that in other scenarios, specificity is encoded in the on-rate. Therefore, it is important to explore how specificity is encoded within this mechanism. Our results indicate that the variation in the dissociation constant $k_{\mathrm{o}\mathrm{f}\mathrm{f}}/k_{\mathrm{o}\mathrm{n}}$ is dominated by $k_{\mathrm{o}\mathrm{f}\mathrm{f}}$, suggesting that specificity is encoded in the off-rate. The magnitudes and variations of these rates could be directly compared to experimental results in future studies.

Figure 4

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The Rouse Model describes a polymer as a chain of point masses connected by springs and characterized by a statistical segment length $l_0$ resulting from the rigidity of springs $k$ and the thermal noise level $k_{B}T$, $l_0^2=3k_{B}T/k$ (Prince, 1953; Doi and Edwards, 1988). We coarse-grain the TF by representing it with a few binding sites for the IDR, and an additional site for the DBD, as illustrated in Figure 1b. The targets are assumed to be positioned at equal spacing, with one target for the DBD and a few targets for the IDR, corresponding to the orange and black squares in Figure 1c. We have also verified that relaxing this assumption, and using randomly positioned targets, does not affect our results (Appendix 6. Robustness of the binding design principles). In the Rouse Model, the 3D probability density describing the relative distance in the equilibrium state, denoted as $r$, between any two sites can be derived through Gaussian integration (Doi and Edwards, 1988), leading to:

where $l=\sqrt{n_{\mathrm{s}\mathrm{e}\mathrm{g}}}{l}_0$ and $n_{\mathrm{s}\mathrm{e}\mathrm{g}}$ is the number of segments between the two sites.

Each binding site of the TF at position ${\mathit{r}}_i$ follows over-damped dynamics, which reads:

where the random variable $\xi$ follows the standard normal distribution. $D$ is the diffusion coefficient of the coarse-grained binding site. The force acting on each site is $\mathit{f}(\left\{{\mathit{r}}_i\right\})\equiv -\mathrm{\partial }{U}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}/\mathrm{\partial }{\mathit{r}}_i$, and

where ${\mathit{R}}_j$ are the positions of the targets. $w_d=1\mathrm{b}\mathrm{p}$ is the typical width of the short-range interaction between sites and their corresponding targets. The typical distance $l_0$ between coarse-grained binding sites is $5\mathrm{b}\mathrm{p}$. The stiffness of the springs is thus $3k_{B}T/l_0^2=3k_{B}T/25\mathrm{b}{\mathrm{p}}^2$. To speed up the simulations, when the sites’ minimal distance to the antenna is larger than the threshold $10\mathrm{b}\mathrm{p}$, the TF is regarded as a rigid body, and all sites move in sync with their center of mass. We verified that the resulting search time is insensitive to the value of this threshold. When the sites’ minimal distance to the antenna is less than $10\mathrm{b}\mathrm{p}$, we consider the excluded volume effect to ensure that different binding sites are not trapped at the same target. Specifically, we use the WCA interaction (Weeks et al., 1971):

where $\sigma =1\mathrm{b}\mathrm{p}$. In this region, we also adopted an adaptive time step: $\mathrm{\Delta }t/(\mathrm{b}{\mathrm{p}}^2/D)=min\left\{1,\frac{k_{B}T}{max|{\mathit{f}}_i|\mathrm{b}\mathrm{p}}\right\}\times 0.01$. Note that this choice ensures that the displacement of the sites at every step is much smaller than the binding site size $w_d$. Due to the IDR trapped in the binding targets on the antenna, which results in slow dynamics, the simulation time becomes significantly longer, as found in previous simulation studies (Vuzman and Levy, 2010; Marcovitz and Levy, 2013). This slowdown is similar to the behavior observed in glass-forming liquids (Berthier and Biroli, 2011). As a result, the individual simulation time steps range from 10⁹ to 10¹¹.

We verify the broad applicability of Equation 6 in the main text, regardless of the specific details of microscopic interactions. To demonstrate this, we only require that the density function of the end-to-end distance $r$ for a segment containing $m$ amino acids (AAs), denoted as $f(r,l_0)$, complies with Equation 6. Specifically, it is.

where $l_0$ is the typical length of the segment. When $m$ is large, according to the central limit theorem, $r$ should conform to the distribution in Equation 10 with a variance of $l_0^2=m{a}_{\mathrm{e}\mathrm{f}\mathrm{f}}^2$, where $a_{\mathrm{e}\mathrm{f}\mathrm{f}}$ represents the effective length of an individual AA.

Next, we will consider a specific form of interaction, of springs with a finite rest length. In one limit (where the rest length vanishes), this corresponds to the Rouse model (where Equation 10 is exact for any value of $m$, not only asymptotically) and in another limit (where the springs are very stiff) this corresponds to a chain of rods. As we shall see, Equation 10 provides an excellent approximation already for moderate values of $m$.

To proceed, we consider two adjacent AAs at positions ${\overrightarrow{r}}_i$ and ${\overrightarrow{r}}_{i+1}$ with an interaction denoted by $u(|{\overrightarrow{r}}_i-{\overrightarrow{r}}_{i+1}|)$. Their probability density is $\rho ({\overrightarrow{r}}_i,{\overrightarrow{r}}_{i+1})\propto \exp (-u(|{\overrightarrow{r}}_i-{\overrightarrow{r}}_{i+1}|)/(k_{B}T))$. $a_{\mathrm{e}\mathrm{f}\mathrm{f}}^2$ can be calculated using $\rho ({\overrightarrow{r}}_i,{\overrightarrow{r}}_{i+1})$:

Specifically, for a spring-like interaction with a stiffness constant $k_0$ and a rest length $b_0$, the interaction is given by $u=\frac{1}{2}{k}_0{(|{\overrightarrow{r}}_i-{\overrightarrow{r}}_{i+1}|-b_0)}^2$. This formula can also be interpreted as a harmonic approximation of the interaction around the equilibrium distance $b_0$. In this case, $a_{\mathrm{e}\mathrm{f}\mathrm{f}}^2$ is derived analytically:

where $a_0^2\equiv 3k_0/k_{B}T$ and $\mathrm{E}\mathrm{r}\mathrm{f}(z)=\frac{2}{\sqrt{\pi }}{\int }_0^z\exp (-x^2/2)dx$ is the error function. When $b_0=0$, we have $a_{\mathrm{e}\mathrm{f}\mathrm{f}}=a_0$, and Equation 10holds for any $m$. When $b_0\gg a_0$, then $a_{\mathrm{e}\mathrm{f}\mathrm{f}}\approx b_0$ corresponds to a chain of rods where the angle between consecutive rods is free to rotate.

Appendix 1—figure 1a shows that the numerical results obtained from this harmonic interaction are in good agreement with Equation 10 even at moderate values of $m$. The inset displays the variation of $a_{\mathrm{e}\mathrm{f}\mathrm{f}}^2/a_0^2$ with $b_0/a_0$ obtained from Equation 12. In Appendix 1—figure 1b, with $b_0=a_0$, $f(r,l_0)$ rapidly converges to Equation 10.

Appendix 1—figure 1

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In this section, we derive the generalized binding probability $P_{\mathrm{T}\mathrm{F}}$ for $n_b$ IDR binding sites and $n_t$ IDR targets. To achieve this generalization, we consider all the possible number of bindings, $n$ from 0 to $min(n_b,n_t)$, and all possibilities for binding orientations for each $n$. Note that the energy and entropy trade-off is naturally included in our model, and the generalized formula of $P_{\mathrm{T}\mathrm{F}}$ is also applicable for non-uniformly distributed binding targets and binding sites.

Based on the Rouse model, the system’s energy can be expressed as $\frac{k}{2}\sum _{i=1}^{n_b}({\overrightarrow{r}}_i-{\overrightarrow{r}}_{i-1}{)}^2=\frac{3k_{B}T}{2l_0^2}\sum _{i=1}^{n_b}({\overrightarrow{r}}_i-{\overrightarrow{r}}_{i-1}{)}^2$, where ${\overrightarrow{r}}_i$ represents the position of the binding site $i$. The probability density in the absence of binding reads:

where $\int d^3{\overrightarrow{r}}_0...\int d^3{\overrightarrow{r}}_{n_b}\rho ({\overrightarrow{r}}_0,...,{\overrightarrow{r}}_{n_b})=1$.

Now we consider a given configuration with $n$ IDR sites bound, where IDR sites $q_1<...<q_n$ are bound to targets $s_1,...,s_n.$ The positions of these bound targets are at ${\overrightarrow{R}}_{s_1},...,{\overrightarrow{R}}_{s_n}$. The DBD site is denoted by $q_0=0$ and its target is denoted by $s_0=0$. There are $n_b-n$ unbound IDR sites, indicated by ${\overline{q}}_1<...<{\overline{q}}_{n_b-n}$. The corresponding binding probability is given by:

where ‘$\mathrm{o}\mathrm{r}\mathrm{n}$’ stands for the orientation of the given configuration, $Z$ is the normalization factor, and $E_{\mathrm{D}\mathrm{B}\mathrm{D}}$ is the DBD site binding energy for its target. There are $(\genfrac{}{}{0ex}{}{n_b}{n})(\genfrac{}{}{0ex}{}{n_t}{n})n!$ possible binding orientations at $n$. The key insight in evaluating this integral is to realize that we can use the result of Equation 6 of the main text to integrate out the degrees of freedom associated with the sites between $q_{i-1}$ and $q_i$:

Since $r_{q_{i-1}}$ and $r_{q_i}$ are confined to a very small volume $V_1$, to an excellent approximation we may simply evaluate $f\left(\left|{\overrightarrow{r}}_{q_i}-{\overrightarrow{r}}_{q_{i-1}}\right|,\sqrt{q_i-q_{i-1}}{l}_0\right)$ in Equation 15 at ${\overrightarrow{r}}_{q_{i-1}}={\overrightarrow{R}}_{s_{i-1}}$ and ${\overrightarrow{r}}_{q_i}={\overrightarrow{R}}_{s_i}$, to obtain:

where $r_{\mathrm{D}\mathrm{B}\mathrm{D}}\equiv \frac{V_1}{V_c}{e}^{E_{DBD}}$, $r_{\mathrm{o}\mathrm{r}\mathrm{n},i}=\left|s_i-s_{i-1}\right|d$, $l_{\mathrm{o}\mathrm{r}\mathrm{n},i}=\sqrt{q_i-q_{i-1}}{l}_0$, and $\mathcal{P}\left(E_B,r_{\mathrm{o}\mathrm{r}\mathrm{n},i}\right)=e^{E_B}{V}_{1}f\left(r_{\mathrm{o}\mathrm{r}\mathrm{n},i},l_{\mathrm{o}\mathrm{r}\mathrm{n},i}\right)$.

To give a specific example, consider the binding topology corresponding to Appendix 2—figure 1. In this case, $n=3$, $l_{\mathrm{o}\mathrm{r}\mathrm{n},1}=l_0$ (since there is one spring between the DBD and the first binding site), $l_{\mathrm{o}\mathrm{r}\mathrm{n},2}=\sqrt{3}{l}_0$ (since there are three springs between $q_1$ and $q_2$) and similarly $l_{\mathrm{o}\mathrm{r}\mathrm{n},3}=l_0$. Equation 15 implies that the contribution of this diagram to the binding probability is: $\frac{1}{Z}\frac{V_1^4}{V_c}\frac{27}{{\left(2\pi l_0^2\right)}^{9/2}}{e}^{3E_B+E_{\mathrm{D}\mathrm{B}\mathrm{D}}-5d^2/l_0^2}$. Note that while in this example, the order of the target sites is monotonic (i.e. as we increase the index of the binding sites, we progressively move further away from the DBD), in principle, we should sum over all diagrams, including those where the order is non-monotonic. At $n=0$, $P_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}(\mathrm{o}\mathrm{r}\mathrm{n},0)=\frac{1}{Z}{r}_{\mathrm{D}\mathrm{B}\mathrm{D}}$.

Appendix 2—figure 1

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Similarly, for the state where the DBD is unbound and $n\ge 2$ IDR sites are bound, the binding probability reads:

where $r_{\mathrm{I}\mathrm{D}\mathrm{R}}\equiv \frac{V_1}{V_c}{e}^{E_B}$. For $n=1$, $P_{\mathrm{u}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}(\mathrm{o}\mathrm{r}\mathrm{n},1)=\frac{n_t{n}_b}{Z}{r}_{\mathrm{I}\mathrm{D}\mathrm{R}}$. Here $n_t{n}_b$ represents the number of possibilities for one of the $n_b$ IDR sites bound to one of the $n_t$ targets. For $n=0$, $P_{\mathrm{u}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}(\mathrm{o}\mathrm{r}\mathrm{n},0)=\frac{1}{Z}\prod _{i=0}^{n_b}{\int }_{{\overrightarrow{r}}_i\notin V_1}{d}^3{\overrightarrow{r}}_i\rho ({\overrightarrow{r}}_0,...,{\overrightarrow{r}}_{n_b})\simeq \frac{1}{Z}$.

Since the sum of all the possibilities is equal to 1, as given by: $\sum _{n=0}^{n_{min}}\sum _{\mathrm{o}\mathrm{r}\mathrm{n}}(P_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}+P_{\mathrm{u}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}})=1$, where $n_{min}=min\{n_b,n_t\}$, we can determine the normalization factor. Therefore, the binding probability $P_{\mathrm{T}\mathrm{F}}$ for the DBD bound in its target can be expressed as:

When $P_{\mathrm{T}\mathrm{F}}\ll 1$, we can approximate $P_{\mathrm{T}\mathrm{F}}$ as

where we have considered the facts that $r_{\mathrm{I}\mathrm{D}\mathrm{R}}<r_{\mathrm{D}\mathrm{B}\mathrm{D}}$ because $E_B<E_{\mathrm{D}\mathrm{B}\mathrm{D}}$, and $P_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}=r_{\mathrm{D}\mathrm{B}\mathrm{D}}/(1+r_{\mathrm{D}\mathrm{B}\mathrm{D}})\approx r_{\mathrm{D}\mathrm{B}\mathrm{D}}$ due to $r_{\mathrm{D}\mathrm{B}\mathrm{D}}\ll 1$. For instance, taking $E_{\mathrm{D}\mathrm{B}\mathrm{D}}=15$, $V_1=1{\mathrm{b}\mathrm{p}}^3\approx (0.34\mathrm{n}\mathrm{m}{)}^3$, and $V_c=1\mu m^3$, we find that $r_{\mathrm{D}\mathrm{B}\mathrm{D}}\sim {10}^{-4}$.

At $n_b=n_t=1$, the generalized Equation 19 simplifies to the following expression:

Therefore, $P_{\mathrm{T}\mathrm{F}}$ can be approximated as $(1+\mathcal{P})P_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}$, as illustrated in Equation 1 of the main text. This approximation is also derivable from Equation 20.

Appendix 5—figure 1 shows the contour lines at $P_{\mathrm{T}\mathrm{F}}=0.9$ and $P_{\mathrm{T}\mathrm{F}}=0.5$ for various $E_B$. All take an approximate symmetrical L-shape, indicating that for a given $E_B$, the optimal values $n_b^{\ast }$ and $n_t^{\ast }$ are comparable, as guided by the black dashed line $y=x$. As $E_B$ increases, the contour lines shift towards lower values of $n_b$ and $n_t$, indicating a decrease in $n_b^{\ast }$ as $E_B$ increases.

Appendix 5—figure 1

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In the main text, we employed uniform binding site distances in TFs, represented as $l_0$, and uniform binding target distances on the DNA, denoted as $d$, to obtain the design principles. Here, we also explore the scenario where both IDR binding sites and their targets are distributed as a Poisson process (i.e. the distance between neighboring sites is exponentially distributed), with mean values of $l_0$ and $d$, respectively. Appendix 6—figure 1 is similar to Figure 2 in the main text. We observe that the design principles identified in the main text are applicable under these conditions as well.

Appendix 6—figure 1

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We show a typical example of the trajectory of a TF (red) in the cell nucleus (light blue) in Appendix 7—figure 1a where the TF walks along the antenna to the DBD target once it attaches the IDR targets at ${\tilde{n}}^{\ast }$ and $E_B^{\ast }$. The DBD target is at the origin. In Appendix 7—figure 1b, we find that the average number of rounds the TF hits the antenna, $n_{\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{s}}$, is close to 1, so it is fair to describe the overall search process as two distinct steps with the first step being purely a 3D search and the second a purely 1D search. This is expressed by Equation 5 in the main text.

Appendix 7—figure 1

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The minimal search time, $t_{min}$, is obtained at $L^{\ast }$, $E_B^{\ast }$, and ${\tilde{n}}^{\ast }$. Appendix 7—figure 2a shows $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ vs. $E_B$ at fixed $L^{\ast }$ and ${\tilde{n}}^{\ast }$. When $E_B>E_B^{\ast }$, $t_{3\mathrm{D}}$ does not depend on $E_B$ as we predict (red solid line) and $t_{1\mathrm{D}}$ exponentially increases with increasing $E_B$. The solid blue line shows the exponential dependence of $t_{1\mathrm{D}}$ on $E_B$, also shown in the inset. When $E_B<E_B^{\ast }$, the number of rounds of hits on the antenna, $n_{\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{s}}$, is larger than 1, as displayed in the inset. In this scenario, Equation 5 in the main text becomes ineffective, resulting in a significantly longer search time than $t_{min}$.

Appendix 7—figure 2b shows $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ vs. $\tilde{n}$ at fixed $L^{\ast }$ and $E_B^{\ast }$. When $\tilde{n}>{\tilde{n}}^{\ast }$, $t_{3\mathrm{D}}$ and $t_{1\mathrm{D}}$ agree with the prediction of Equation 5 (red curve and blue curve). Interestingly, we find that $t_{1\mathrm{D}}$ only weakly depends on $\tilde{n}$. This arises due to the compensating effects of attaching and detaching: As $\tilde{n}$ increases, the TF attaches to new targets more easily, speeding up 1D motion. However, it also makes detaching from the current site more difficult, thereby slowing down 1D motion.

When $\tilde{n}<{\tilde{n}}^{\ast }$, Equation 5 breaks down because of multiple rounds of hits, which is verified in the inset. As a result, both $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ and $t_{3\mathrm{D}}$ increases with decreasing $\tilde{n}$, leading them to become significantly longer than $t_{min}$.

If each round of 3D search is independent, the 3D search time should be $n_{\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{s}}$ times the one round of 3D search time, displayed in the black curve.

Appendix 7—figure 2

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In this section, we will show that the optimal search process (i.e. minimizing the mean first passage time) generically takes place via a single round of 3D-1D search. To do so, we first need to derive the expression for the mean total search time, $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, and its dependence on the relevant parameters, including, importantly, $E_B$ and $L$. According to our model presented in the main text: A TF is initially in the 3D nucleus space; it can attach to the antenna and perform 1D diffusion with the coefficient denoted by $D_1$. We assume the DBD target is placed in the middle of the antenna, as shown in Figure 1 of the main text. The mean search time taken for a TF to move from 3D space and attach to the antenna is denoted by $t_{3\mathrm{D}}$. The TF can also detach from the antenna at a rate we denote by $\mathrm{\Gamma }$, perform a 3D random walk, until ultimately attaching again to the antenna. In the following, we will make the assumption (often made in the context of FD model Berg et al., 1981; Berg and von Hippel, 1987; Slutsky and Mirny, 2004; Mirny et al., 2009; Li et al., 2009; Bénichou et al., 2011; Sheinman et al., 2012) that the binding position after such a 3D excursion is uniformly distributed on the antenna.

Both $D_1$ and $\mathrm{\Gamma }$ are functions of $E_B$. When the TF is on the antenna, its mean search time to reach the DBD target, located a distance $x$ away on the antenna, is represented by $\mathcal{T}(x)$. Naturally, $\mathcal{T}(x)$ is zero when $x=0$, the location of the DBD target. The mean total search time $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ is thus given by

Consider a time interval $\mathrm{\Delta }t$. With probability $\mathrm{\Gamma }\mathrm{\Delta }t$ the TF falls off, and the mean first passage time to reach the target is $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}+\mathrm{\Delta }t$. Otherwise, with probability $1-\mathrm{\Gamma }\mathrm{\Delta }t$, the mean first passage time would be the average of the two possible outcomes, namely, $(\mathcal{T}(x+\mathrm{\Delta }x)+\mathcal{T}(x-\mathrm{\Delta }x))/2+\mathrm{\Delta }t$. This recursive reasoning leads us to the following equation:

In the continuous limit, we have

where $D_1\equiv (\mathrm{\Delta }x{)}^2/(2\mathrm{\Delta }t)$. By applying the boundary conditions, specifically the absorbing boundary condition at $x=0$ and the reflecting boundary conditions at $x=\pm L/2$, $\mathcal{T}(x)$ is determined as

where $L_s$, representing the 1D diffusion distance, is defined by $L_s\equiv 2\sqrt{D_1{\mathrm{\Gamma }}^{-1}}$(Wunderlich and Mirny, 2008). Therefore, $t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ can be self-consistently solved by substituting $\mathcal{T}(x)$ into Equation 22, presented as follows: