2.1 Contact space representation of polymer topology

We propose a theoretical framework to model a compact polymer domain of length L as a network in contact space. In this model, each bead (or node) of the polymer represents a coarse-grained chromatin segment of size 1 kbp. The conformation of the polymer is encoded in a matrix C_ij of dimension  +  ×  +  , where . Two beads i and j are assumed to be in contact with a probability , where and . The polymer topology enforces for all i. The exponent γ serves as a measure for polymer compaction. Initially, we assume the domain to be a uniform network by setting , leading to a uniform contact probability . Here, p_u effectively quantifies the compaction of the domain: a low p_u represents a 1D chain, whereas corresponds to a fully connected network. For a broader perspective, we extend the model to networks with power-law contact probabilities for varying values of the contact probability exponent γ. As the polymer collapses, contacts become more frequent, resulting in smaller values of γ. For a self-avoiding polymer, , which decreases to for a random walk polymer. Chromosomes, on average, exhibit [39], while experiments within highly folded TADs report even smaller γ [34]. We construct power-law networks that follow for different γ values, closely mimicking polymer domains with different levels of compaction. Unless stated otherwise, we use c = 0.0214, consistent with experimentally observed distributions of c. The polymer connectivity depends both on the number of intersegmental contacts, and on their distribution (γ), and these together determine the compactness of the polymer domain. In general, for similar numbers of intersegmental contacts, a larger exponent implies longer loops, and hence more compact domains. For both the uniform and power-law networks, we interpret our results in terms of the polymer compaction, although this is achieved only through varying the average number of contacts in the uniform case, and both the number and their distribution for power-law networks.

2.2 Protein as random walker

In this network representation, a protein is modelled as an unbiased random walker navigating within a polymer domain ( − 1]). At each time step (τ), a protein located at bead i can either slide along the chain to one of its 1D neighbours (i  +  1 and i–1) or hop to a genomically distant (but spatially adjacent in 3D) bead j if C_ij = 1 (Fig 1b, 1c). Additionally, proteins can unbind from the polymer with a probability , entering the bulk phase. While in the bulk, the protein explores the solution phase with a mean exploration time of [49] before reattaching to the polymer. We assume the exploration time is large enough that the memory of the unbinding site is lost, and the protein rebinds randomly to any site of the polymer, consistent with previous studies [49,50]. When the protein reaches either boundary site, it is considered to have successfully reached the target.

2.3 Static and dynamic polymer domains

We consider two scenarios based on whether the connectivity matrix evolves during the target search process. If the polymer configuration (C_ij) remains constant over time, the target search occurs on a static polymer configuration. However, chromatin organisation within a cell is now widely recognised as a dynamic structure rather than a static one [51]. This dynamic behaviour arises from various factors, including tethering of chromatin to the nuclear lamina, interactions with diverse proteins, and the dynamic process of loop extrusion mediated by cohesin [52]. If C_ij evolves with time, the target search takes place on a dynamic polymer. Below, we outline the procedure for generating the connectivity matrix and the simulation methodology for both scenarios.

2.4 Theoretical calculation of search times

We compute the configuration-averaged mean search times , defined as the time a protein takes to reach either boundaries of the polymer for the first time. The mean search time can be computed as

where is the time taken to reach the target starting from the i^th site. The matrices and are defined by,

(1)

where, τ is the time taken for a single step, C_ij denotes the connectivity matrix, and N_i denotes the total number of bonds for the i^th site. The site i = f denotes the freely diffusing state in solution. For a single configuration, the variable C_ij takes the value 1 with probability P_c(s = |i−j|), given by

(2)

Eqs 1 and 2 can be solved to obtain estimates of the search time to find the target.

2.5 Polymer models of chromatin

In order to validate our results obtained from the contact space representation, we simulate search on topologies obtained from three polymer models - the freely rotating chain, the Lennard-Jones bead spring polymer, and a soft Lennard-Jones polymer.

2.5.2 LJ bead-spring polymer model.

We simulate a bead-spring polymer consisting of L beads each of size (diameter) σ. The total energy (E) of the polymer is given by-

Here is the position vector of i^th bead. The first term corresponds to the energy contribution from harmonic springs ( − ) that connect adjacent beads along the chain. The second term describes attractive interactions governed by the Lennard-Jones (LJ) potential:

Here, ϵ denotes the strength of the attractive interaction.

2.5.3 Soft LJ bead-spring polymer model.

For the soft LJ polymer, the energy (E_soft) is given by [28]:

The expression V₀ − ϵ denotes the energy penalty associated with complete overlap, and the parameters and are utilised to adjust the softness of the potential, as illustrated in S12 Fig. For r < r_m, the potential is repulsive but softer than the standard LJ potential. For , the potential transitions to the LJ form, inducing attractive interactions. Beyond the cut-off distance (), the interaction energy is set to zero. The parameters used in this study are: , variable, , r_m = 1.12, .

Simulation of bead-spring polymer.

The Langevin dynamics simulations were performed using the LAMMPS software package [56]. In this approach, the position of each polymer bead evolves according to the equation:

where r_i is the position of the center of mass of the i-th bead, m_i is its mass, E_i is the sum of all the interaction potentials acting on bead i. The friction coefficient determines the diffusion constant . For polymer beads, we set m_i = 1 and . is a noise term with components which satisfy: − , where is component α of the noise vector for object i, and and are the Kronecker and Dirac delta functions respectively. These equations are solved using a velocity-Verlet algorithm with time step , where τ is the simulation time unit defined by .

2.5.4 Simulation of protein walk on polymer.

We transform simulated polymer structures with varying compaction levels into network representations (C_ij). This transformation is achieved by defining a cut-off radius (r_c), which determines contact between non-bonded beads. Specifically, if the 3D distance (r_ij) between two non-bonded beads (i,j) in the polymer structure is less than or equal to the cut-off radius, they are considered in contact, and C_ij is set to 1. Otherwise, C_ij is set to 0:

We also calculate total number of non-neighbouring bonds (N_b) for a given configuration, which is defined as,

Using these network representations, we perform random walks and measure the time required to reach target. The results are averaged over 10,000 independent runs each across 1,000 different configurations to compute .

2.6 Construction of TADs from experimental Hi-C data

To study the protein search process within realistic chromatin architecture, we used chromatin conformation capture data from Hi-C experiments. Our analysis was based on a high-resolution 1 kb Hi-C map derived from the GM-12878 human lymphoblastoid cell line [42]. Specifically, we utilised KR-normalised [57] Hi-C contact frequency matrices f_ij for all autosomes (Chr1–Chr22). We then determine the maximum value of the non-diagonal () contact frequency for each chromosome and divide each f_ij by this number to derive a contact probability map (P_ij) where all elements are scaled between 0 and 1 [58]. Furthermore, our analysis of topologically associating domains (TADs) relied on a curated set of 8,355 TADs from [42], annotated using the Arrowhead algorithm.

Method-1: We define intra-TAD contact probability P_c(s) of two points separated by genomic distance s as the average contact probability between of all genomic loci separated by genomic distance s within a TAD. Contact probability exhibited a power-law, within a range of values of s. We measured γ as the slope of the best-fit line and c as the y-intercept, when plotted on log-log axes, within a chosen range of distances. The obtained γ value signifies the compaction of the corresponding TAD. We calculate the average search time () for each TAD using our theoretical framework with c and γ as input.

Method-2: To simulate the dynamics of a protein within a realistic chromatin structure, we employ a previously described model based on sliding and intersegmental jumping on a network. This network replicates the chromatin architecture and interactions within a single TAD in 3D space. The network is constructed using P_ij values derived from experimental Hi-C data as follows:

where r_n is a uniform random number in the range [0,1]. Here, C_ij represents the connectivity of the chromatin polymer within a TAD. In addition to the connectivity derived from P_ij using Hi-C data, we enforce to maintain polymer connectivity along the backbone. After constructing the network based on Hi-C data, we simulate protein dynamics within this chromatin network. The ensemble average search time () is calculated by averaging over 1,000 distinct configurations, with 10,000 search simulated in each configuration.

3.1 Search time varies non-monotonically with compaction of domain

Motivated by experimental results of very high estimates of time a protein spends non-specifically bound to the DNA [11,59–62], and consistent with high volume fractions inside TADs (S1 Text, S1 Fig) [63], we first consider the limiting case (see Sect 2), so that target search happens only along the polymer, via 1D sliding and intersegmental jumps. For simplicity, we first assume , so that all beads have a uniform connection probability, . We observe a non-monotonic relation between and p_u (Fig 2a). When p_u = 0, the protein performs a 1D random walk, giving where [64]. As p_u increases, more paths to reach the boundaries emerge, reducing . However, beyond a threshold connectivity, increases due to excess intersegmental bonds diverting the walker. This non-monotonic behaviour of is independent of whether the polymer network is static (Fig 2a black symbols) or dynamic (Fig 2a red symbols, and S2 Figa). For dynamic polymer networks, we obtain an exact analytic expression for the search time as a function of the connection probability (See S2 Text). Dynamic rewiring of network connections reduces , with faster configuration changes yielding lower at low p_u (S2 Figa). However, at high p_u values, network dynamicity offers little advantage, as most nodes already have numerous long-range connections. Systematically increasing the switching time relative to the protein sliding timescale allows us to smoothly interpolate between the dynamic and static results (S2 Figa). At low p_u, to leading order, we have,

(3)

a decreasing function of p_u (see S2 Text). At high connection probability, as , all sites become equivalent, and a simplified governing equation can be solved to yield (see S2 Text),

(4)

which implies a linear increase with p_u as , also consistent with a solution using the effective medium approach [65] (see S2 Text, S5 Fig). As can be seen from Fig 2a inset, the analytic expression at both small and high p_u limits matches the simulation results. The optimal behaviour of is consistent for differential rates of sliding and intersegmental jumps (S2 Figb), for all polymer lengths (S3 Fig) and different initial protein positions (S4 Fig). Also, while we have assumed that the target site at the boundary is insulated from the interior sites motivated by earlier studies [32,66], our central result is independent of this strong insulation, and persists when the protein can hop directly to the target site with low (but finite) rates (S6 Fig). Note that in the two extreme limits, this uniform network construction does not represent valid polymer conformations. As , there are no intersegmental bonds, implying a pure rod-like conformation. In the opposite limit , all monomers are connected to all other monomers, which is not possible for physical polymers. Hence the search times in these limits may not be physical. Further, for real polymers, the contact probability scales with the genomic separation. Despite these assumptions, this simple analytically tractable model allows to capture the non-monotonic behaviour of search times with increasing compaction, and hence, intersegmental jumps.

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Fig 2. Search times with 1D sliding and intersegmental hops are minimised for an optimal polymer compaction.

(a) Scaled mean search time for static (black) and dynamic (red) conformations in a uniformly connected network domain vs probability of contact (p_u) between any two non-neighbouring beads for L = 500. The solid blue line is the exact analytical solution of Eqs 1 and 2 (see also S2 Text). The inset shows the low-p_u and high-p_u approximations from Eqs 3 and 4. (b) Scaled mean search time in power-law connected network domain as a function of contact probability exponent(γ) for L = 500 also shows non-monotonic behaviour for both static (black) and dynamic (red) configurations. The blue line is the theoretical calculation from Eqs 1 and 2. (c) For open conformations (low p_u or, high γ), the search is dominated by 1D sliding (blue segments), with very few intersegmental hops, resulting in slow target search. For collapsed conformations (high p_u or, low γ), due to many intersegmental hops (red segments), search times are high. At optimal compactions, balanced 1D sliding and intersegmental hops leads to an optimisation of search times.

https://doi.org/10.1371/journal.pcbi.1013843.g002

Having established this central result in the case , we now turn to the more realistic scenario in which we introduce intersegmental hops with a power-law connectivity, (see Sect 2). For a self-avoiding polymer (SAW), , which reduces to for a random walk (RW). For chromosomes, on average, [39], while within highly folded TADs, experiments report even smaller values for γ [40]. Analysis of Hi-C data inside single TADs have reported contact probability exponents as low as [67]. As a polymer undergoes collapse, contacts become more frequent, reducing the contact probability exponent γ. We explore the behaviour of search times across the full biologically and physically relevant range of . We observe a non-monotonic behaviour of the search times for both and static and dynamic realisations of such power-law networks, as shown in Fig 2b. Again, when the polymer is relatively open (larger γ), dynamic rewiring of the connectivities helps reduce search times, similar to the observation from the simpler uniform connectivity () case. As the polymer compacts from a self-avoiding conformation, search times can decrease by an order of magnitude, followed by a rise of up to five times at very low γ compared to the minimum search time. Note that the minimum γ occurs in the regime , suggesting a possible connection with observed compaction in folded TAD domains. The precise location of the optimum γ value depends on the choice of c. We verify that varying c within biologically relevant ranges does not qualitatively affect our result (S7 Fig). Further note that the relative magnitudes of the decrease and subsequent increase of search times are smaller in this case compared to the uniform probability case. The power-law contact probability takes into account realistic polymer properties and is hence more reflective of search times that can be achieved in biological contexts.

This central result of the importance of polymer conformations is shown schematically in Fig 2c, along with illustrative simulation trajectories. For relatively open conformations, the kinetics is dominated by 1D sliding (blue segments), with very few intersegmental hops, resulting in large target search times. As the polymer becomes increasingly compact, distant segments come into contact, and these intersegmental hops lead to faster search. Conversely, for highly collapsed conformations, there is a preponderance of long hops (red segments), again increasing search times. In between these two extremes lies a regime of optimal compaction, where a judicious mix of 1D sliding and intersegmental hops leads to an optimisation of search times.

3.2 Role of bulk diffusion in target search within compact domains

The classical facilitated diffusion model posits that target search happens through repeated rounds of 1D sliding and bulk 3D diffusion, with an optimal unbinding rate that minimises search times. However, this formalism ignores the role of intersegmental jumps due to the polymer topology itself. We now ask whether bulk diffusion is an advantageous search strategy when we take into account the effects of the polymer compaction itself (see S2 Text). For a protein searching for targets through a combination of 1D sliding and intersegmental hops, the polymer compaction allows a route to minimize search times. How is this search strategy affected if we now consider the effect of bulk 3D diffusion, in addition to 1D sliding and intersegmental jumps? Can search times be further decreased by introducing bulk excursions? Our contact space representation of a polymer allows us to take into account the role of 3D diffusion (), with the assumption that rebinding times are long enough that the memory of the unbinding site is lost, and rebinding occurs randomly at any site along the polymer [49,50]. Experimental estimates of the 1D diffusion coefficients of proteins are in the range of 10⁴ − . Unbinding rates for various proteins have also been reported in the range 10–2000s⁻¹. Taken together, these imply unbinding probabilities for the searching protein in the range 10⁻¹–10⁻⁴ for kinetics of TFs [11,15,68].

We first consider the case of (Fig 3a), so that . As , we recover the classical facilitated diffusion result of an optimum search at an intermediate value of . This is also shown in the cross-section plot for in Fig 3c where results in minimum search times. As p_u increases, the advantage obtained through bulk diffusion reduces, until beyond a certain threshold, unbinding from the backbone offers no advantage to the target search process. The regime of validity of this standard 1D+3D facilitated diffusion is surprisingly small, with only a few intersegmental bonds ( for L = 500) enough to disrupt this classical result. Beyond this threshold, within this random rebinding model, 3D unbinding is not advantageous for target search, with the search times increasing monotonically with increasing in this regime. This can also be seen from the cross-section plot for , showing bulk diffusion is not advantageous for compact domains (Fig 3c). Interestingly, our result of an optimal compaction minimising search times continues to hold true even in the presence of 3D diffusion, with being minimised at an optimal connectivity (p_u). This result holds true for all unbinding probabilities below , consistent with estimated physiological unbinding probabilities.

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Fig 3. Optimal polymer compaction minimises search times even in presence of 3D diffusion.

Contour plot of as a function of 3D unbinding probability () and polymer compaction for (a) a uniformly connected domain and (b) a power-law connected domain of L = 500. In a relatively open chain (see left side in (a) and right side in (b), there is an optimal for which is minimum suggesting 3D excursion is advantageous for these conformations (red curves in (c) and (d)). However, as the polymer collapses, monotonically decreases with decreasing suggesting that 3D excursions are not advantageous for compact polymers (blue curves in (c) and (d)). For biologically relevant regimes, there always exists an optimal polymer compaction where is minimum. The fact that the global minimum of occurs for highly folded conformations and in regions where unbinding is low suggests the role of 3D spatial structure of chromatin in the search process. In (e), trajectory of protein for open polymer configurations (low p_u or, high γ) shows frequent detachment to bulk from the polymer and reattachment to the polymer. In (f), trajectory of protein for polymer at optimal compaction shows rare unbinding from the polymer and frequent intersegmental jumps.

https://doi.org/10.1371/journal.pcbi.1013843.g003

This general result holds true when we look at power-law connectivities (Fig 3b) as well. For the self-avoiding polymer (), 3D diffusion offers an advantage to target search. As the polymer collapses, and hence γ decreases, the relative speed-up through bulk excursions decreases. Beyond the RW polymer (), unbinding from the backbone offers no advantage to the target search process, as intersegmental hops allow a more efficient mechanism to explore distant sites. Indeed for collapsed polymers (low γ or large p_u), and in the vicinity of the optimal compaction state, remarkably, 3D diffusion is not helpful for target search, with a monotonic increase in search times with increasing unbinding probability (S8 Fig). This contrasting effect of bulk diffusion depending on the polymer configuration can also be seen in the cross-section plots for an open () vs compact ( polymer in Fig 3d, showing a monotonic increase with increasing 3D unbinding for the compact polymer (S8 Fig). This central result is shown schematically in Fig 3e and 3f. Our qualitative results continue to hold true for a wide range of bulk residence times, (S9 Fig). For low compaction, bulk excursions are frequent, interspersed with 1D sliding, while for optimal compaction, there are very few bulk excursions, with a mixture of sliding and intersegmental hops dominating the kinetics of the search process.

3.3 Polymer models of chromatin identify the existence of optimal compaction

We now examine whether non-monotonic search times persist when canonical polymer models are used to model chromatin structure (see Sect 2). In contrast to the contact space representation of polymers, in this case the allowed intersegmental jumps arise directly from polymer topology, and hence random walks on these backbones have direct relevance to protein search processes. As polymers collapse, the contact probability changes, and this is reflected in a coupled change of both the pre-factor c and the exponent γ. Thus search processes on polymer backbones offer a pathway to study the effect of compaction on search times distinct from the contact space representation of polymers. A variety of equilibrium as well as non-equilibrium polymer models have been studied in the literature to understand chromatin organisation at different scales [34,69]. We restrict ourselves to three canonical polymer models - the freely rotating chain, the bead spring Lennard Jones polymer, and a soft Lennard Jones polymer. Having investigated the role of bulk diffusion for compact domains, we limit ourselves to target search by 1D diffusion and intersegmental hops on these polymer backbones.

We first study the freely rotating chain (FRC) model (see Sect 2.5.1) which has been used to investigate chromatin behaviour [53,54]. The FRC angle θ (Fig 4a) controls the compaction of the polymer, with large θ leading to a collapsed state (Fig 4b and S10 Figa). This simple model does not take into account excluded volume or other interactions. Nevertheless this can act as controlled model system from where to begin our investigation into the role of compaction from polymer topologies. The FRC bond angle introduces a persistence length l_p which decreases with increasing θ. The power-law scaling of the contact probability holds only for lengths above the persistent length, . Further, the contact probability exponent does not depend on the FRC angle, instead the change in contact probability may be interpreted in terms of an increase in the prefactor c with increasing θ (see S10 Figc). As the polymer collapses, the number of allowed intersegmental jumps increases. We quantify the number of such allowed intersegmental contacts N_b, which increases monotonically with θ (see S10 Figb). We simulated search on these FRC backbones, and the mean search time shows a non-monotonic behaviour with increasing bond angle, and hence polymer compactness (Fig 4c), and also the number of intersegmental bonds (Fig 4c inset). Thus in line with the expectations from the network model of polymer, there exists an optimal compaction where search times are minimum.

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Fig 4. Search on polymer backbones reveals the existence of an optimal configuration that minimises search time.

(a) Schematic of a FRC polymer (see Sect 2.5.1) with a bond angle θ. (b) Representative configurations of FRC polymers of length L = 500 at , showing increasing compaction with increasing θ. (c) in FRC polymers as a function of bond angle θ, and the scaled number of non-bonded contacts (inset) for L = 200–500. − L denotes the maximum possible number of non-bonded contacts. (d) Schematic of a bead-spring polymer with attractive LJ interactions between non-bonded pairs (see Sect 2.5.2). (e) Representative configurations for polymers of length L = 500 at LJ interaction strengths (ϵ) = 0.1,0.4,1.2. Lower ϵ results in open conformations with higher R_g and lower non-bonded contacts, N_b. Stronger ϵ lead to compact, globule-like structures with lower R_g and higher N_b. (f) in LJ polymers as a function of interaction strength ϵ and the scaled number of non-bonded contacts (inset) for L = 200–500. (g) In polymers with soft LJ interactions, bead overlap is permitted and controlled by a softness parameter (see Sect 2.5.3). Soft LJ polymers exhibit more compact configurations at higher ϵ compared to regular LJ polymers. (h) Representative configurations of soft LJ polymers of length L = 500 at . (i) in soft LJ polymers as a function of ϵ and (inset) for L = 500, shown for different softness levels characterised by .

https://doi.org/10.1371/journal.pcbi.1013843.g004

We next use a bead-spring polymer model (Fig 4d) of chromatin with attractive Lennard-Jones (LJ) potential (see Sect 2.5.2) [70–72]. This introduces self-avoidance effects absent in the FRC model. With increase of the LJ interaction strength ϵ, the polymer undergoes a second-order phase transition from a coiled to a compact state (Fig 4e, and S11 Figa), accompanied by an increase in the number of possible intersegmental jumps (see S11 Figb). As the polymer collapses, the scaling exponent γ reduces from the self-avoiding exponent. Beyond the collapse transition, γ does not decrease further, even with increasing ϵ. For these compact conformations, the power-law description itself only holds at intermediate scales, with a saturation for very long range contacts [73,74] (see S11 Figc). The search times on these LJ polymer backbones exhibited a non-monotonic pattern as the LJ interaction strength is increased (Fig 4f), or equivalently as the connectivity of the polymer increases (Fig 4f inset). There was a distinct minimum observed in the search times, occurring close to physiological chromatin compactions. Thus a LJ polymer qualitatively mirrors the results obtained from the uniform network and FRC polymer models.

However, unlike the significant rise seen in the FRC and the network model, the extent of increase beyond a critical ϵ is less pronounced in this case. This limited increase is due to the inherent hard-core repulsion in the LJ potential, which imposes a lower limit on polymer compaction, and on the number of intersegmental bonds. Since γ saturates beyond the critical ϵ, this increase can also be interpreted in terms of a marginal increase in the total number of contacts at the same γ. This interplay between LJ interactions and polymer structure thus introduces a unique feature that distinguishes it from both FRC and the network model.

Is this weak non-monotonic rise in then the correct physical description of search times? The answer hinges on whether the hardcore LJ potential best represents coarse-grained chromatin conformations. Experimental and theoretical studies of chromatin configurations and 3D distances suggest that a soft inter-bead potential that allows for overlap is a more realistic description of coarse-grained chromatin [28,75]. We study a polymer model (Fig 4g) with a soft inter-bead potential (see Sect 2.5.3, Fig 4h), with the softness controlled by the parameter (S12 Fig). We investigate on such soft polymers with varying degrees of softness. A lower value of indicates a softer polymer, whereas high resembles the classical LJ potential. For a very soft polymer (), shows a strong non-monotonicity, analogous to the FRC and the network results (Fig 4i). This is because a soft polymer core cannot effectively enforce volume exclusion, resulting in a more tightly packed polymer compared to the conventional LJ model (see S13 Figa). This then leads to a higher number of non-neighbouring contacts within the polymer domain (see S13 Figb) and hence a stronger non-monotonicity in search times (Fig 4i inset). In terms of the contact probability, the initial decrease is then due to the change of the scaling exponent γ. Beyond the collapse transition, the soft potential allows for more monomers to come into contact, increasing the number of possible intersegmental hops, leading to the non-monotonic behaviour (see also S13 Figc). As we decrease the softness (increasing ), the degree of non-monotonicity in the search times decreases and approaches the LJ-like behaviour. Thus non-monotonic behaviour of search times remains a crucial feature of protein diffusion on coarse-grained chromatin-like soft polymer topologies. The degree of non-monotonicity in is then controlled by the softness of the inter-bead potential.

3.4 Topological domains are self-organised near optimal compaction

Is this non-monotonic behaviour of search times relevant for real TADs? Where do TAD conformations lie with respect to this optimal compaction state? How fast is the search process for optimal chromatin compaction? To answer these, we turn to publicly available experimental Hi-C contact probability data [42] and perform the following theoretical study (see Sect 2). We identify 8355 TADs across all chromosomes from the GM12878 cell line (see Fig 5a and S14 Figb) and analyse Hi-C data of these TADs at 1 kbp resolution. To characterise each TAD’s spatial organisation, we quantify the contact probability as a function of genomic separation, P_c(s) obtained from the normalised Hi-C frequency matrix. The identified TADs have a broad distribution of lengths ranging from 65 kbp - 2600 kbp (see S14 Figc). For each TAD of genomic length L, we fit the contact probability curve to a power-law over a range s = 10 to (L–100) kbp, as shown in Fig 5b. This allows us to quantify two parameters - the contact probability scaling exponent γ, and the prefactor c, for each individual TAD. We then analyse the distributions of the fitted values of c and γ across all TADs, as shown in Fig 5c and 5d, which yield average values of and . Our estimate of is also consistent with values which reproduce experimentally known mean spatial distances between genomic loci [28]. Both the contact probability exponent and the prefactor are independent of the size of the TAD (see S14 Figd). The most probable values of γ fall within the range of approximately 0.66 to 0.75, indicating a consistent degree of compaction among these TADs, and implying that topological domains are more compacted than the overall chromosome organisation, which has an average contact probability exponent [39]. We first compute protein target search times across a broad spectrum of domain lengths L and contact probability exponent values γ on a power-law network with c fixed at its average value, . The resulting search time estimates are represented as a contour map in Fig 5e in the (L,γ) phase plane. To assess how these findings relate to optimal search efficiency in TADs, we overlay the fitted γ values and corresponding lengths L for each individual TAD onto the contour map in Fig 5e, as indicated by red dots. Strikingly, we observe that the TAD conformations cluster near regions of the contour map corresponding to minimal search times, irrespective of the length of the TAD, facilitating rapid target finding and potentially enhancing genomic function. This clustering of TADs near the optimal region is insensitive to variations of the prefactor c over the biologically relevant range (Fig 5d), as is shown in S7 Fig.

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Fig 5. Experimentally identified TADs occupy the zone of optimal compaction and leads to ballistic search.

(a) An example of 3 distinct TAD regions in Hi-C contact map from [42]. (b) Contact probability (P_c(s)) as a function of genomic distance (s), fitted using for the three TADs in (a). (c-d) Distribution of best fit parameters c and γ for all 8355 collected TADs from GM12878 data [42]. (e) Contour plot of scaled mean search times as a function of γ and L. The fact that experimentally observed [42] L and γ values (red dots) fall very close to the minimum is yet another evidence that 3D chromatin structure is relevant in the search processes. (f) Search times as a function of TAD lengths show a very different scaling compared to purely 1D diffusive search (pink dashed line, ). Search time computed from simulating walks on network with Hi-C contact probability (red dots) and network with observed power-law properties (black) show . The observed exponent is very different from the target search time scaling on self avoiding walk (SAW) polymer indicating role of distinct chromatin organisation in search processes. (g) Scaling exponent () of with domain length L for different contact probability scaling exponent γ. is higher for both open () and compact polymer () configuration and shows a minimum near the experimentally observed γ range of TAD, marked by green star.

https://doi.org/10.1371/journal.pcbi.1013843.g005

To further investigate the impact of chromatin organisation at the TAD scale on target search, we now turn to the question of how fast is the target search process on these optimally compacted chromatin conformations? To address this, we generate chromatin networks by employing two distinct methodologies (see Sect 2.6 for details). First, we construct power-law networks for each Topologically Associating Domain (TAD) of length L using parameters c and γ from fitting the power-law distribution to Hi-C data. This method allows us to simulate chromatin networks with averaged contact probabilities, and hence spatial relationships, which mimic the conformations of real topological domains in chromatin without requiring detailed region-specific contacts. In the second approach, we created direct Hi-C networks based on experimentally measured contact probabilities P_ij between chromatin loci i and j within each TAD (S14 Figa). Unlike the power-law method, this approach incorporates region specific contact probabilities obtained from Hi-C data, thus preserving detailed sequence-specific interactions within chromatin. This method allows us to capture the unique spatial organisation within each TAD and explore how these specific contacts impact the search dynamics. The mean search times on these networks are shown in Fig 5f. Interestingly, we find that the search times scale with TAD lengths in a near-ballistic manner, following . This near-ballistic scaling suggests that the organisation within these TADs enables target search to proceed in an effectively driven fashion, even though the underlying dynamics is purely diffusive. For pure one-dimensional sliding along the polymer contour, one would expect . However, intersegmental hops change this scaling of search times to yield a contact probability dependent scaling, . This effective exponent also shows a minimum around the TAD range (Fig 5g), with larger values both for collapsed globule, and random walk or self-avoiding polymers. Our findings imply that random walks on these power-law-structured TADs can significantly accelerate search processes, offering a large speed-up relative to purely diffusive timescales (where ). This efficiency appears to result from the inherent structure of TADs rather than from any additional motor or energy-dependent mechanisms, as illustrated in Fig 5f. Moreover, we find that this result is consistent in both the approaches - power-law networks, and Hi-C based contacts, suggesting that although Hi-C probability-based networks capture detailed region-specific effects, in the context of search processes, power-law models offer an accurate approximation of search times expected for biological search processes on chromatin. Thus, these models provide a robust framework for estimating protein dynamics and highlight the importance of TAD organisation in optimising genomic search efficiency.

4.1 Strengths and limitations of the model

Our theoretical analysis, for the first time, investigates the role of chromatin conformations itself as a control parameter in regulating search times. The strength of our analysis lies in the fact that we use experimental contact probability data from Hi-C experiments to understand how intersegmental contacts at the TAD scales shapes search times. Although there has been significant attention on the connection between gene expression and enhancer-promoter interactions facilitated by chromatin looping, the role of the broader 3D chromatin architecture on protein search remains largely unexplored. By dissecting the sub-TAD organisation, we aim to bridge the gap in understanding how chromatin’s 3D structural features contribute to the efficiency of protein search processes within the genome. While our analysis focuses on relatively short genomic regions (up to 2 Mbp), the methodology we employ is quite general and can be extended to larger scales. However, our contact space network representation of a polymer makes simplifying assumptions about the polymer conformations. In particular it neglects higher order correlation due to the looped structures [77]. Such correlations are not expected to play a major role for dynamic polymers [44,78], assuming that the search timescales are larger than the timescales of polymer dynamics, and this allows us to use this representation to calculate search times. Further, using Hi-C data also has its inherent limitations. Hi-C contact matrices represent time-averaged contact frequencies generated from millions of reads, and how this translates to real chromatin conformation is still an open question. Hi-C contact matrices also measure relative probabilities, and does not determine the prefactor in the scaling relation. However, our choice of the prefactor c has been shown to yield sensible estimates of 3D distances between genomic loci [28], and hence are expected to correspond to biologically relevant ranges. Finally, while our analysis focuses solely on the role of the chromatin conformation, other biological factors may also influence target search times. These include the role of epigenetic modifications, as well as dynamic processes such as loop extrusion, nucleosomal kinetics and breathing which affect local polymer properties [79]. However, the timescales of protein search, which is much faster than loop extrusion [80], and the 1 kbp resolution, which is larger than nucleosomal lengthscales [28], allows us to ignore these dynamic processes in our analysis. Note that our 1 kbp coarse-graining also implies that for bulk diffusion, we neglect very fast local rebinding, as these are absorbed into the effective residence time on the coarse-grained beads. We also neglect any cooperative effects or interactions between proteins, or recruitment of transcription factors due to interactions with other protein machineries, such as within transcription factories. Nevertheless, our analysis highlights how regulating polymer conformations alone can result in faster search by a factor of 10–50 compared to target search on extended conformations.

4.2 Suggestions for new experiments to test our predictions

Our key prediction is that chromatin structure affects target search. To test this prediction, one can generate DNA or chromatin configurations with different structures having different compactions. One possible way to achieve is this is to insert multiple artificial CTCF binding sequences in the genome so that more looped structures are formed. One can engineer (either within the genome or in a long plasmid outside the chromosomes) the location and direction of these boundary sequences such that a 3D structure of a given contact probability exponent γ may be achieved. One can measure the statistics of protein reaching a well-defined sequence-specific target on this structure. Another way to assemble chromatin structure of varying compaction is to assemble DNA/chromatin with varying concentrations of DNA-compacting proteins like HP1 or other architectural proteins [81–83], as well as salt concentration to tune detachment events in vitro [76]. Systematic experiments can then be designed to explore how these changes affect protein dynamics and hence search times. Recent advances in single molecule tracking experiments also allow us to study the dynamics of TFs over long time scales [84]. Visualisation of a pluripotency factor Sox2 inside a topological domain observed distinct hopping behaviour which suggested that chromatin domains sequester proteins and kinetically facilitates search, consistent with the results of our analysis. Systematic tracking of TF motion in different topological domains with varying levels of compaction using such single particle techniques can further help validate our hypothesis.

4.3 Outlook and relevance to chromatin organisation and function

The hierarchical organisation of chromatin in mammalian cells, from chromosome territories at scales of Mbp to topologically associating domains at hundreds of kilobases has now been recognised from various experiments [42]. This scale dependent organisation implies that search strategies by proteins may also need to be suitably modified at these different scales. While bulk diffusion in the nucleoplasm can be effective in locating the right spatial domain for a particular protein, target search at the scale of TADs requires leveraging chromatin contacts into intersegmental hops. Crucially, intersegmental transfers, which depend on the topology of the polymer, and hence introduces correlated jumps offers a distinct mechanism from bulk diffusion. Indeed, our analysis shows that search processes which leverage allowed intersegmental hops are generally much faster than those that proceed through a combination of 1D sliding and 3D diffusion. Target search inside the nucleus through intersegmental transfers has consequences not only for gene expression via transcription factor binding, but also plays a role in DNA damage repair, where repair proteins need to locates the site of DNA breaks [85]. Our findings highlight the critical role chromatin structure plays in influencing the search dynamics of nuclear proteins, and is consistent with emerging experimental data from a range of diverse experiments [76,84,86].