Unfolding of the chromatin fiber driven by overexpression of noninteracting bridging factors

Abstract

Nuclear molecules control the functional properties of the chromatin fiber by shaping its morphological properties. The biophysical mechanisms controlling how bridging molecules compactify chromatin are a matter of debate. On the one side, bridging molecules could cross-link faraway sites and fold the fiber through the formation of loops. Interacting bridging molecules could also mediate long-range attractions by first tagging different locations of the fiber and then undergoing microphase separation. Using a coarse-grained model and Monte Carlo simulations, we study the conditions leading to compact configurations both for interacting and noninteracting bridging molecules. In the second case, we report on an unfolding transition at high densities of the bridging molecules. We clarify how this transition, which disappears for interacting bridging molecules, is universal and controlled by entropic terms. In general, chains are more compact in the case of interacting bridging molecules because interactions are not valence limited. However, this result is conditional on the ability of our simulation methodology to relax the system toward its ground state. In particular, we clarify how, unless using reaction dynamics that change the length of a loop in a single step, the system is prone to remain trapped in metastable, compact configurations featuring long loops.

Significance

In the cell nucleus, the chromatin fiber is organized by a myriad of nuclear molecules in a way that is specific to the cell phenotype. Bridging molecules can pack chromatin by mediating interactions between reactive sites of the fiber. We study how the thermodynamic parameters of the bridging molecules affect the chromatin conformation. Intriguingly, an excess of bridging molecules does not necessarily further compactify the fiber; rather, it could swell it. We also highlight the impact of different reaction dynamics between bridging molecules and the reactive sites of the fiber. Our results clarify how thermodynamic and kinetic details at the molecular scale drastically affect the larger-scale structure of the chromatin fiber.

Introduction

Nuclear DNA in eukaryotic cells is scaffolded by histones and other proteins forming a fiber known as chromatin. A myriad of molecules, mainly proteins and ribonucleic acids (RNAs), regulate the morphological properties of chromatin by selectively tagging and bridging specific loci of the genome (1, 2, 3, 4). Different conformations of the chromatin fiber result in different functional states. Only a small fraction of nuclear DNA does not directly interact with regulatory factors (5). The correlations between regulatory factors, the fiber’s morphology, and the system’s functional properties are a matter of debate.

Coarse-grained models have been employed to study the three-dimensional structure of chromatin. A typical class of models uses bead-and-spring chains (see Fig. 1), in which monomers (which may represent a segment of chromatin containing multiple histones) interact through selective interactions (6, 7, 8, 9, 10), which could change as a result of chemical reactions triggered by epigenetic regulators (11). A bottom-up parameterization of specific systems is often out of reach, given the lack of an understanding of how different regulators precisely interact. Numerical studies linking microscopic interactions to mesoscopic structures may then help to infer molecular interactions. For instance, (12) used coarse-grained simulations to show how DNA-bound ParB proteins should polymerize and simultaneously be capable of forming three-dimensional bridges to prevent fragmentation of DNA-bound ParB condensates. Bead-and-spring models are also used to reconstruct the most likely conformation of the fiber compatible with experimental results in chromosome-capture techniques (13) measuring the Euclidean proximity between different chromatin loci (as expressed by connectivity maps) (14,15).

Figure 1.

(a) Noninteracting bridging molecules can bind tagged sites of the fiber (in gray) and drive compaction of chromatin through the formation of loops. (b) For high densities of the bridging molecules, each reactive monomer carries a B molecule, loops open, and chromatin unfolds. (c) Microphase separation of interacting B molecules can drive compaction of the fiber.

Chromatin conformation below the megabase scale is often regulated by bridging factors (e.g., HP1, YY1, CTCF, …) cross-linking reactive segments. We study the morphological properties (compaction, connectivity map, and loop size distribution) of chromatin fibers cross-linked by bridging factors (see Fig. 1 a). We study how the concentration of the bridging molecules (ρ_B) and the affinity between bridging molecules and tagged (or reactive) monomers (quantified by the association constant K_a) affect the phase behavior of the system. Intriguingly, as foreseen in (16), for high values of ρ_B (and a given value of K_a), the fiber unfolds given that all reactive monomers carry a linker and bridges cannot form (see Fig. 1 b). As compared to previous studies (e.g., (17,18)), in our model, each reactive bead can interact with at most a single bridging factor. We discuss the effect of relaxing this constraint in the Results and discussion. We rationalize simulation results using correspondence states identified by a nondimensional parameter (ϕ). The overexpression of a bridging factor can affect the phenotype (e.g., (19)). However, we are not aware of molecular studies of the chromatin fiber at high values of ρ_B.

The unfolding transition of Fig. 1 b is somehow analogous to the resolubilization of virus crystals (20), proteins (21), and nucleic acids (22,23) at a concentration of multivalent ions (forming intermolecular bridges) higher than some millimoles. Similarly, aggregates of ligand-presenting colloids cross-linked by short DNA oligomers dispersed in solution melt when increasing the concentration of the bridging molecule beyond a threshold value (ranging between 10⁻⁸ and 10⁻⁴ M) that depends on the number of ligands per particle (24).

Simulation approaches are challenged by the multiscale nature of the system (25, 26, 27), as well by the necessity of sampling the large variability of the possible fiber configurations. In this work, we address the latter issue and deploy an ensemble of Monte Carlo (MC) moves that allow changing the connectivity state of a chain in a single step by binding/unbinding two monomers while simultaneously regrowing a fraction of the chain. The proposed method, along with other simulation strategies implementing reaction moves in systems of multivalent chains (28, 29, 30, 31, 32, 33, 34, 35, 36), are ideal for studying the unfolding transition given that the high free-energy barriers between competing states are sidestepped by dedicated MC moves. On the other hand, MC methods are not suitable to study the dynamics of the system unless the reaction timescales are much bigger than the backbone relaxation times (33).

Loop formation is not the only way of compactifying chromatin’s segments. Bridging molecules may interact through residual (e.g., multivalent) interactions (37,38). Importantly, while remaining soluble in the solution, the bridging molecules may condensate on the chromatin fiber and form finite-sized drops enveloping reactive monomers (see Fig. 1 c; (16)). The condensation of bridging molecules results in effective interactions between reactive monomers, which are not valence limited, finally driving the folding of the fiber (17). With interacting bridging molecules, we find that the reentrant transition disappears. In general, interacting bridging molecules lead to more compact configurations. This result only holds when comparing equilibrated states. When not changing the length of the loops with dedicated MC moves (which do not affect the equilibrium distribution of the system), chains folded by noninteracting bridges become more compact. The results and methodology presented in this work allow assessing the thermodynamics of competing mechanisms leading to domain formation in chromatin.

Methods

The coarse-grained model

Fig. 2 presents the model. We consider fully flexible freely jointed chains made by N monomers ({r} = {r₁, r₂, …, r_N}) with bond length equal to σ. We do not assume any particular short-range organization of the nucleosomes (affecting local properties like the persistence length). Given the generality of the result, we employ a prototypical model rather than focusing on a specific system. N_R monomers out of N can reversibly bind bridging (B) molecules dispersed in solution. μ_B and ρ_B are, respectively, the chemical potential and the density of the B molecules in solution. We evenly distribute the reactive (R) monomers along the chain and define the degree of functionalization as f = N_R/N. ν₁ and ν₂ are, respectively, the list of reactive monomers carrying B molecules and the list of pairs of monomers cross-linked (see Fig. 2). N₁ and N₂ are then the number of R monomers carrying a factor and not forming a loop and the number of B molecules cross-linking two reactive monomers. We model the atomistic details of the reactive monomers (when free, carrying a linker, and cross-linked) using internal partition functions (respectively, q_R, q_RB, and $q_{R_{2}B}$; see Fig. 2 a). The use of implicit partition functions allows parametrizing the model using the experimental association constant (K_a; Fig. 2 a) and tuning the magnitude of the reaction rates. This feature is not directly available in models using explicit representations of the potential between reactive units but requires fine-tuning procedures. Steric repulsions between monomers are modeled through repulsive pair interactions (u^R). When considering interacting bridging molecules (IBMs), monomers carrying linkers interact through pair potentials (u^A) that include an attractive well. Notice that we model bridging molecules implicitly. In particular, the attractive part of u^A accounts for indirect interactions mediated by the clustering of bridging molecules (see Fig. 1 c). In this work, u^R and u^A are represented by two cut-and-shifted Lennard Jones potentials

$$u^X\left(r\right)=\{\begin{array}{cc}4\left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right]-4\left[{\left(\frac{\sigma }{{\lambda }_X}\right)}^{12}-{\left(\frac{\sigma }{{\lambda }_X}\right)}^6\right]& r<{\lambda }_X\\ 0& r>{\lambda }_X\end{array},$$ (1)

with X = A, R and λ_R = 2^1/6σ, λ_A = 2.5σ. The partition function providing the statistical weight of each microstate reads as follows (if β = 1/k_BT where k_B is the Boltzmann’s constant):

$$Z=\sum _{{\nu }_1,{\nu }_2}{\left(q_R\right)}^{N_R-N_1-2N_2}{\left(q_{RB}\right)}^{N_1}{\left(\frac{q_{R_{2}B}}{{\text{Ω}}_0}\right)}^{N_2}{e}^{\beta {\mu }_B\left(N_1+N_2\right)}{\mathcal{Z}}_{{\nu }_1,{\nu }_2},$$ (2)

$${\mathcal{Z}}_{{\nu }_1,{\nu }_2}=\int \text{d}\left\{r\right\}{e}^{-\beta U_{{\nu }_1{\nu }_2}\left(\left\{r\right\}\right)}\prod _{\left\{i,j\right\}\in {\nu }_2}\delta \left(|{\mathbf{r}}_i-{\mathbf{r}}_j|-\sigma \right),$$ (3)

where $U_{{\nu }_1{\nu }_2}=\sum _{i,j}{u}_{ij}\left(\left|{\mathbf{r}}_i-{\mathbf{r}}_j\right|\right)$ and u_ij = u^R or u^A as discussed above. ${\mathcal{Z}}_{{\nu }_1,{\nu }_2}$ is the contribution to the partition function accounting for the possible configurations of the chain backbone at a given ν₁, ν₂. A pair of monomers (i and j) cross-linked by a bridging molecule ({i, j} ∈ ν₂) are constrained to stay at a fixed distance equal to σ. In Eq. 3, the integral over the chain’s backbone is constrained by the fixed bond length condition. Using the language of (36), Eqs. 2 and 3 define a probability distribution on a stratification: a hierarchy of nested manifolds in 2(N − 1) − N₂ + 3 dimensions (where 3 refers to the center-of-mass degrees of freedom).

Figure 2.

(a) Formation of dimers (RB) and trimers (R₂B) starting from reacting units (B molecules and R monomers) in diluted conditions. (b) B molecules bind chains carrying R monomers and lead to the formation of loops. ν₁ is the list of R monomers forming an RB complex and ν₂ the list of pairs of monomers cross-linked by B molecules. ν₂ can be represented using connectivity maps that, when averaged, provide the likelihood of finding two monomers cross-linked by a B molecule.

Linking the internal partition functions to the association constant

As highlighted in a thread of investigations developing reaction ensemble Monte Carlo methods (39,40), q_X (X = R, B, RB, and R₂B) are linked to the association equilibrium constant (K_a) measured in diluted mixtures of B molecules and R monomers, the latter not constrained to stay on a chain (Fig. 2 a). Specifically, the partition function of a single molecule of type X is V × q_X, where V is the volume of the system corresponding to the configurational space accessible to the center of mass of X. In diluted conditions, we can calculate the chemical potential (μ_X) from the partition function of N_X molecules as follows:

$$Z_X\left(N_X\right)=\frac{{\left(V{q}_X\right)}^{N_X}}{{\sigma }_X{N}_X!}\Rightarrow {\mu }_X=k_{B}T\text{log}\frac{{\rho }_X{\sigma }_X}{q_X},$$ (4)

where ρ_X is the density of the molecule X and σ_X the symmetry order of the coarse-grained representation of X (σ_R = σ_RB = 1, ${\sigma }_{R_{2}B}$ = 2, and σ_B = 1 as we treat B molecules implicitly). We now consider the two reactions leading to the formation of RB and R₂B molecules: R + B $\rightleftharpoons$ RB and RB + R $\rightleftharpoons$ R₂B. In equilibrium conditions, the sum of the chemical potentials of the educts should be equal to the same quantity calculated for the products: μ_R + μ_B = μ_RB and μ_R + μ_RB = ${\mu }_{R_{2}B}$. When using Eq. 4, these equations allow expressing the internal partition functions in term of the association constant K_a as follows:

$$\frac{q_{RB}}{q_R{q}_B}=\frac{{\rho }_{RB}}{{\rho }_R{\rho }_B}\equiv K_a\frac{q_{R_{2}B}}{q_{RB}{q}_R}=2\frac{{\rho }_{R_{2}B}}{{\rho }_{RB}{\rho }_R}\equiv 2K_a,$$ (5)

where we assume that the two terminals of the B molecules react with the same strength when binding the first and the second R monomer. When considering reactive monomers tethered to a chain, we can use Eq. 5 in Eq. 2 to calculate the statistical weight $\left({\pi }_{N{\text{′}}_1,N{\text{′}}_2}\right)$ of binding a B molecule to the chain (N₁ → N₁ + 1) and of forming a loop (N₁ → N₁ −1 and N₂ → N₂ + 1) relative to the one of a reference state $\left({\pi }_{N_1,N_2}\right)$:

$$\begin{gathered} \frac{{\pi }_{N{\text{′}}_1=N_1+1,N{\text{′}}_2=N_2}}{{\pi }_{N_1,N_2}}={\rho }_B{K}_a\frac{{\mathcal{Z}}^{\text{′}}}{\mathcal{Z}} \\ \frac{{\pi }_{N{\text{′}}_1=N_1-1,N{\text{′}}_2=N_2+1}}{{\pi }_{N_1,N_2}}=2\frac{K_a}{{\text{Ω}}_0}\frac{{\mathcal{Z}}^{\text{′}}}{\mathcal{Z}} \end{gathered}$$ (6)

The previous expressions clarify how ρ_B and K_a are the only parameters required to parameterize the model. Such quantities are usually experimentally accessible. Instead, the configurational contributions $\left(\mathcal{Z}\text{'}/\mathcal{Z}\right)$ are sampled using Monte Carlo simulations of the coarse-grained model, as detailed in the next section. Ω₀ (see Eqs. 2 and 6) is the configurational volume available to the orientational degree of freedom of an R₂B dumbbell (Ω₀ = 4πσ²). In Eq. 2, we divide $q_{R_{2}B}$ by Ω₀, given that both $\mathcal{Z}$ (Eq. 3) and $q_{R_{2}B}$ (Eq. 4) account for all possible orientations of an R₂B molecule. Notice that Ω₀ does not divide q_RB in Eq. 2 because Z does not depend on the orientation of the B molecules attached to the chain.

The simulation strategy

First, we consider the MC moves that change the connectivity state of the chain ν₂ by reacting complementary monomers. When attempting to form a new loop (for instance, by reacting monomers i with j in Fig. 3 a), we start from deleting from the system a fraction of the chain (Γ) in the proximity of j (dashed segment in the top of Fig. 3 a). We then generate a new segment configuration $\left({\text{Γ}}^{R_{2}B}\right)$ with i linked with j (Fig. 3 a, bottom). This process is done as proposed in (35) using methods growing chains with fixed endpoints (41, 42, 43, 44). In the reverse move, we delete a segment of the chain $\left({\text{Γ}}^{R_{2}B}\right)$ containing a randomly selected monomer forming a loop (j) and regrow a segment (Γ) without any loop. Acceptance rates are calculated as done in configurational bias Monte Carlo (45, 46, 47). Importantly, we calculate the relative statistical weights of the two configurations of Fig. 3 a using the second of Eq. 6. The acceptance rates become then a function of the association constant K_a. We refer to Supporting materials and methods, Section S2.3 for the details of the algorithm.

Figure 3.

(a) Loop/unloop MC moves react complementary monomers (i and j) while updating a segment of the chain encompassing j (highlighted using dotted line). (b) MC moves involving multiple reaction events or changing the number of B molecules attached to the chain are shown.

We also used MC moves that implement multiple binding and unbinding events simultaneously. In the swap MC move (Fig. 3 b), monomer k detaches from i and sequentially binds a second complementary monomer j (35). In the swing MC move (Fig. 3 b), two pairs of reacted monomers exchange their partners (35). In this work, we also use an MC move in which two reactive monomers carrying a B molecule react while simultaneously detaching one of the two bridges (loop/unloop + unbind/bind in Fig. 3 b). B molecules are reversibly attached to the chain using the binding or unbinding moves (Fig. 3 b). Contrary to the swing and the swap move, the binding and unbinding moves do not update the configuration of the backbone {r}. The details of the MC moves of Fig. 3 b are reported in Supporting materials and methods, Section S2.2 (bind/unbind) and Supporting materials and methods, Sections S2.4, S2.5, and S2.6 (for the loop/unloop + binding/unbinding, the swap, and the swing, respectively).

To further relax the system, we also employ standard MC moves that update the backbone of the polymer without affecting ν₁ and ν₂. Specifically, we use pivot and double pivot moves that rotate fractions of the chains. Segments of the chain are also regrowth using the CBMC method (as used in the loop/unloop move) without changing the connectivity state of the system. A dedicated MC move displaces and reorients reacted complexes (R₂B) while regrowing a fraction of the surrounding network. In each MC cycle, we randomly perform one of the previous moves.

Comparison with existing methodologies implementing supramolecular reactions

Simulation schemes to study the statistical properties of chains carrying reactive units are finding applications in nanotechnology, for instance, to self-assemble polymeric nanoparticles (48). Molecular dynamics simulations have been used to study the morphological properties of the nanoparticles with an emphasis on finding protocols leading to maximal chain compaction (48,49). In most of these studies, cross-linking between complexes is irreversible. (32,50) introduced a three-body potential that allows implementing swap-like moves in molecular dynamics simulations. Recently, (36) introduced a general MC scheme that allows adding or removing holonomic constraints reversibly. Intermolecular, reversible linkers are also used to enforce topological entanglement when studying polymer melts using soft potentials (51). In this respect, we note that in this version of our simulation method, the chains are crossable. On the one hand, the crossability of the chromatin fiber is guaranteed by the action of dedicated enzymes. Secondly, topological constraints could be included in our methodology using linking numbers.

Chains carrying monomers featuring selective interactions are currently used to study condensation of multivalent proteins into membraneless bodies (28, 29, 30). These contributions study the molecular determinants underlying the aggregation of multivalent proteins comprising short, folded domains linked by intrinsically disordered regions. In particular, the LASSI package (30) employs lattice models in which folded domains are mapped into beads forming reversible linkages and disordered regions into strings of beads interacting through nonspecific interactions. The lattice model is parametrized by atomistic simulations and is sequence dependent (52). Relevant to the findings of this work, nonspecific interactions between linkers could lead to microphase separation in multicomponent systems (29) and affect the physical mechanism underlying chain aggregation (28). The MC moves presented in the previous section could be readily adapted to the LASSI setting. Finally, simulations based on field-theoretic methods (53) have also been used to study supramolecular polymer physics (31).

Results and Discussion

Noninteracting bridging factors: monofunctional chains

In this section, we quantify how the density of bridging factors (ρ_B) and the association strength between proteins and the fiber (K_a) regulate the number of reactive monomers carrying linkers (N₁) as well as the number of loops (N₂). We then correlate N₁ and N₂ with the morphological properties of the chains, namely the gyration radius, the distribution of the loop length, and the averaged connectivity map. When forbidding the formation of loops, the bindings of bridging proteins to different reactive monomers become independent events, and the probability that a reactive monomer carries a B molecule becomes equal to (see the discussion after Eq. 8)

$$\varphi =\frac{{⟨N_1⟩}_{\text{noloops}}}{N_R}=\frac{K_a{\rho }_B}{1+K_a{\rho }_B},$$ (7)

where ${\langle \cdots \rangle }_{\text{noloops}}$ denotes a constrained average with N₂ = 0 as obtained using Eq. 2. Motivated by this observation, in the following, we use the variable ϕ to discuss our results. For a given bridging protein, ϕ is usually experimentally accessible and could be used to compare the results of the two prototypical systems considered in this study (IBMs and NIBM; see Fig. 1) with staining experiments and, finally, infer the microscopic interactions between bridging factors.

Figs. 4 a and Document S1. Supporting materials and methods and Figs. S1–S12, Document S2. Article plus supporting material a study the fraction of reactive monomers carrying a linker without forming a loop ($\langle N_1\rangle$/N_R) as a function of ϕ when changing K_a (Fig. 4 a) and f (Document S1. Supporting materials and methods and Figs. S1–S12, Document S2. Article plus supporting material a). As expected, $\langle N_1\rangle$ increases with ϕ from N₁ = 0 (for ϕ = 0) to N₁ = N_R (for ϕ = 1). When increasing the association constant, $\langle N_1\rangle$ starts to deviate from the linear behavior predicted by Eq. 7 because of loop formation. In particular, at a given ϕ, loops become more favorable at high values of K_a given that the formation of loops at a given N₁ + N₂ is only controlled by the association constant and free-energy terms (discussed below), which do not depend on ρ_B (see Eq. 2). In Figs. 4 b and S1 b, we study the number of loops as a function of ϕ. As anticipated above, $\langle N_2\rangle$ increases with K_a (Fig. 6 b) and the degree of functionalization f (if f ≤ 0.5; see Fig. S1 b). Importantly, the number of loops is nonmonotonic in ϕ and goes to zero when ϕ tends toward ϕ = 1. This observation underlies the unfolding of the chain when overexpressing bridging factors. Entropic terms regulate the opening of a loop in favor of two R monomers carrying two bridging molecules. The process is purely entropic because the number of reacted monomers (and therefore the number of K_a terms entering the Boltzmann distribution; see Eq. 2) in the two competing microstates does not change. For high values of ρ_B, the system attempts to maximize the number of bridging factors present on the chain, therefore opening loops.

Figure 4.

Fraction of reactive monomers (a) carrying a factor not engaged in a loop, (b) forming a loop, and (c) number of B molecules attached to the chain per R monomer as a function of ϕ. N = 1000 and f = 0.5. In all cases, the error bar is smaller than the size of the symbols.

Figure 6.

(a) Averaged radius of gyration $\langle R_g^2\rangle$ of chains folded by NIBMs as a function of ϕ. $\langle R_{g,0}^2\rangle$ is the averaged gyration radius in a system without B molecules. (b) Loop length distribution for different values of K_a and ϕ is shown. In (a) and (b), we use N = 1000 and f = 0.5 and calculate error bars using 50 independent simulations consisting of 5 × 10⁵ MC cycles. (c) and (d) are snapshots with K_a = 500 when, respectively, using and not using swap and swing MC moves (Fig. 3b). White, gray, and black beads represent, respectively, nonreactive monomers, not-looped R monomers, and looped monomers.

Intriguingly, for all values of K_a and f reported in Figs. 4 b and S1 b, the plots of $\langle N_2\rangle$ as a function of ϕ are symmetric with respect to the axis ϕ = 1/2. To understand this result and make new predictions, in Fig. 5 we present a pathway to estimate the free energy of two systems at different ρ_B (corresponding to a given ϕ and ϕ′ = 1 − ϕ with ϕ′ > ϕ) and same K_a. Taking as a reference state a polymer without any bridging factor, we decompose the free energy of the system into the contribution of putting N₁ + N₂ bridging factors on the chain (ΔF_A) and the one arising from making N₂ loops (ΔF_B) without attaching any extra linker to the polymer. ΔF_A can be calculated as

$$\beta \text{Δ}{F}_A=-\text{log}\left[\left(\begin{array}{l}{N}_R\hfill \\ N_1+N_2\hfill \end{array}\right){\left(K_a{\rho }_B\right)}^{N_1+N_2}\right]$$ (8)

Figure 5.

A two-step pathway to calculate the free energy of the system with a given N₁ and N₂. ΔF_A is the free energy of attaching N₁ + N₂ B molecules to the fiber, and ΔF_B accounts for the loops’ contribution to the free energy.

In Eq. 8, the binomial term counts the ways of distributing N₁ + N₂ factors within N_R R monomers, and K_aρ_B is the statistical weight of each microstate as obtained using Eqs. 2 and 5 (notice that Eq. 7 follows from Eq. 8). Using Stirling’s approximation, we obtain

$$\beta \text{Δ}{F}_A=\left(N_1+N_2\right)\text{log}\frac{N_1+N_2}{N_R}+\left(N_R-N_1-N_2\right)\text{log}\frac{N_R-N_1-N_2}{N_R}-\left(N_1+N_2\right)\text{log}\left(K_a{\rho }_B\right).$$ (9)

The calculation of ΔF_B is not feasible. Mean-field estimates of ΔF_B are also tricky because, in contrast to systems with ligand-presenting colloids, reactions between complementary monomers can hardly be treated as independent events. Even though analytic expressions of this term are not available, we can prove that ΔF_B is the same for the two systems considered in Fig. 5. This claim follows from the fact that, in the two systems, the numbers of empty R monomers and R monomers carrying a bridge are exchanged. It follows that the combinatorial and configurational terms entering the calculation of ΔF_B are the same in the two cases because B factors do not affect the local morphology of the chain. Therefore, in general, we can write the following equality:

$$\text{Δ}{F}_B\left(K_a,N_2,N_1+N_2\right)=\text{Δ}{F}_B\left(K_a,N_2,N_R-N_1-N_2\right)$$ (10)

The previous equation explains the reason why $\langle N_2\rangle$ is symmetric with respect to the axis ϕ = 1/2; at a given value of N₁ + N₂, the two terms of Eq. 10 are minimized by the same value of N₂.

We can push our analysis a step further and predict that the total number of bridging factors on the chain ($\langle N_1+N_2\rangle$) is an antisymmetric function of ϕ. The most likely number of N₁ and N₂ follows from saddle point equations

$$\frac{\partial \left(\text{Δ}{F}_A+\text{Δ}{F}_B\right)}{\partial N_1}=0\frac{\partial \left(\text{Δ}{F}_A+\text{Δ}{F}_B\right)}{\partial N_2}=0$$ (11)

When developing the first of these equations using Eq. 8, we obtain

$$\frac{\partial \left(\text{Δ}{F}_A+\text{Δ}{F}_B\right)}{\partial N_1}=0\iff \text{log}\frac{N_1+N_2}{N_R-N_1-N_2}+\frac{\partial \text{Δ}{F}_B}{\partial N_1}=\text{log}\left(K_a{\rho }_B\right)=\text{log}\frac{\varphi }{1-\varphi },$$ (12)

where we use Eq. 7 to express K_aρ_B in terms of ϕ. Under the transformations N₁ + N₂ → N_R − N₁ − N₂ and ϕ → 1 − ϕ, all the terms of the second equation change sign. It follows that $\langle N_1+N_2\rangle$ is an antisymmetric function of ϕ along the axes ϕ = 1/2 and N₁ + N₂ = N_R/2. We verify this prediction in Figs. 4 c and Document S1. Supporting materials and methods and Figs. S1–S12, Document S2. Article plus supporting material c using simulation results. In particular, the total number of bridging factors on the chain increases with ϕ. Similar to Figs. 4 c and Document S1. Supporting materials and methods and Figs. S1–S12, Document S2. Article plus supporting material c, when increasing the values of K_a and f, the plots deviate from a linear behavior (corresponding to N₂ = 0). For ϕ < 1/2, $\langle N_1+N_2\rangle$ > ϕ because two R monomers can cooperatively stabilize a B molecule through the formation of a loop. For ϕ > 1/2, however, $\langle N_1+N_2\rangle$ < ϕ because multiple B molecules are shared by pairs of R monomers. When ϕ → 0 and ϕ → 1, the behavior of $\langle N_1+N_2\rangle$ is dominated, respectively, by $\langle N_2\rangle$ and N_R − $\langle N_2\rangle$.

Importantly, the previous arguments rely on the fact that we did not consider nonspecific interactions between bridging factors and the chain’s backbone. For instance, in the presence of steric interactions, Eq. 10 (and similarly Eq. 8) would not be valid because more bridges on the chain would increase the configurational cost of folding the fiber. Although we expect the general trend to remain unaffected by nonspecific interactions, the simulation methodology can easily be adapted to account for these terms.

In Fig. 6 a, we study the averaged squared radius of gyration $\langle R_g^2\rangle$ as a function of ϕ for f = 0.5. We observe that $\langle R_g^2\rangle$ is nonmonotonic in ϕ with the chains that reswell when the system tends toward ϕ = 1. This behavior mirrors the trends observed for the number of bridges (see Fig. 4 b) and clarifies how intramolecular linkages drive the compaction of the fiber. In particular, $\langle R_g^2\rangle$ decreases when increasing the value of K_a because more loops become stable. In Fig. 6 b, we report the probability of forming a loop made by L segments for different values of K_a and ϕ. The L = 0 value refers to the probability for an R monomer to be unlooped. Consistently with Fig. 6 a, increasing K_a decreases the probability of finding monomers unpaired. Moreover, the system attempts to minimize the length of loops as a result of the higher configurational cost of forming longer loops. As a result, the loop length distribution sharply decreases with L. Longer loops are expected when reducing the nonspecific repulsion between monomers (u^R in Eq. 1). However, the system can feature persistent longer loops unless employing MC moves that change the length of a loop in a single step (35). This result is shown in Fig. 6, c and d, in which we report two fiber structures when using (Fig. 6 c) or not using (Fig. 6 d) the swap and the swing MC move (Fig. 3 b). In the second case, we obtain a much more compact structure with longer loops that persist during the simulation. Low dissociation rates can lead to arrested structures featuring long loops. Once a long loop forms, it could be further stabilized by dimerization of proximal reacting beads. Instead, the swing and swap moves allow minimizing the loops’ length without unbinding two reacted monomers (see Fig. S2). Notice that short loop configurations maximize configurational entropy and are thermodynamically stable. This observation is consistent with pioneering works on associating polymers (54). Intriguingly, the reaction kinetics has a drastic effect on the system’s conformation. Longer loops will also occur when using semiflexible backbones. The backbone’s rigidity is determined by the number and short-range organization of the nucleosomes represented by a single bead and is not considered by this investigation. However, we stress how the unfolding of the fiber at high ρ_B is not model dependent and will be found in all systems featuring noninteracting bridging factors. As further proof, in Supporting materials and methods, Section S3, we consider a system in which the bridging factors interact differently with two types of reactive monomers (as specified by two association constants). At high bridge concentrations, we recover unfolding transitions, which are not a function of the association constants. Below, we further discuss how to use our result to estimate the unfolding line in systems in which higher-order complexes (e.g., associating YY1 or CTFC) mediate the formation of loops. Consistently with the fact that thermodynamic states only feature short loops, Figs. S3 and S4 show how the results of Figs. 4 and 6 a are not affected by the chain’s length.

IBMs

In this section, we consider the case in which B molecules interact as due, for instance, to multivalent interactions (16). Here, interacting bridging molecules can condensate around multiple R sites that can therefore cluster. Following the strategy of (55), we drive the clustering of R monomers carrying bridging factors using the LJ potentials introduced in Eq. 1. We do not further refine the model because our primary purpose is to warn that the previous section’s reentrant transition disappears in systems with interacting bridging factors.

Fig. 7 a reports the averaged gyration radius of the IBM model as a function of ϕ for different degrees of functionalization f. As compared to Fig. 6, $\langle R_g^2\rangle$ rapidly decreases with ϕ and then exhibits a plateau for ϕ $\underset{\approx }{>}$ 0.1. The sharp decrease of $\langle R_g^2\rangle$ in ϕ is due to a cooperative effect in which B molecules are colocalized by R monomers and stabilized by their mutual interaction. We verify this claim in Fig. S5, in which we compare the fraction of R monomers carrying a B factor in the interacting and noninteracting model. More bridging factors are found on the chain in the case of interacting bridging molecules because in this model, R monomers can simultaneously interact with multiple partners. This observation also explains the fact that chains are more compact in the IBM model (compare Fig. 7 a with Fig. 6 a). Similar findings have been reported in synthetic systems, in which scholars struggled to self-assemble compact polymeric nanoparticles using intramolecular bridges (48).

Figure 7.

(a) Averaged radius of gyration $\langle R_g^2\rangle$ of chains folded by IBMs as a function of ϕ for K_a = 500. $\langle R_{g,0}^2\rangle$ is the averaged gyration radius in a system without bridging molecules. We have calculated error bars using 50 independent simulations consisting of 5 × 10⁵ MC cycles. (b–d) Typical snapshots for K_a = 500 and different f, ϕ are given. White, gray, and black beads represent, respectively, nonreactive monomers, not-looped R monomers, and looped monomers.

Fig. 7 a does not exhibit the unfolding transition, and the chains stay compact even for ϕ = 1 (notice that for ϕ = 1, the system becomes equivalent to a copolymer with attractive beads). This result follows from the fact that monomers carrying B molecules are not passivated but can still interact as a result of the interactions between B molecules. This result is confirmed by snapshots showing how for f = 0.5 and different values of ϕ, the morphology of the fiber is comparable (see Figs. 7, c and d and S6 for the same configurations, in which we only report beads bound to a B molecule). On the other hand, smaller values of f increase the size of the chain as R monomers interact with fewer partners (see Fig. 7, a and b). As compared to the NIBMs, the model of this section leads to longer loops (see Figs. 6 b and S7). This result is a direct consequence of having collapsed chains with an increased probability of finding monomers belonging to different chain sections nearby.

The models of Figs. 6 and 7 consider the extreme cases in which R monomers interact with at most a single partner (Fig. 6) or with every R monomer nearby (Fig. 7). More realistically, R monomers may represent kilobase segments tagged with multiple reactive sites. In this case, the interactions between monomers are not valence limited (but still ρ_B dependent), even for noninteracting bridging factors. However, the fiber will still unfold, as supported by a previous work reporting on the melting of colloidal crystals when increasing the concentration of a bridging factor (24). Our simulation procedure could be easily adapted to a more general model suitable to study specific systems.

Loops mediated by multimeric factors

Multimeric bridges often mediate loop formation. Therefore, it is important to address the generality of the unfolding transition in the presence of more complex reactions. Importantly, we can study the general case starting from chains functionalized by self-complementary reactive beads dimerizing with an equilibrium constant equal to K_eff (34,35). Fig. S8 reports R_g as a function of K_eff for an f = 0.2 system. First, it is possible to map the model with noninteracting bridging molecules into the system of Fig. S8 by writing K_eff as a function of the density of bridging factors ρ_B and the association constant K_a. In particular, K_eff reads as follows: $K_{\text{eff}}^{\left(B\right)}$ = ρ_B × (K_a)²/[1 + ρ_B × K_a]². Using $K_{\text{eff}}^{\left(B\right)}$ and Fig. S8, in Fig. S9 we recover the results of the NIBM model.

We now employ a similar strategy to study looping mediated by factors tagging the fiber (with affinity K_a) and then dimerizing (with affinity K_d). This example is relevant to study, for instance, loops mediated by YY1 (56) or CTFC factors (57). In this case, K_eff reads as follows: $K_{\text{eff}}^{\left(D_2\right)}$ = (ρ_D)² × K_d × K_a/[1 + K_a × ρ_D + K_d × K_a(ρ_D)²]², where ρ_D is the density of the factors. The previous equation can be used to draw the phase behaviors as a function of ρ_D, K_a, and K_d using tabulated data of Fig. S9. This strategy is generally valid to the extent that bridging complexes do not interact. For instance, the previous expression can readily be adapted to systems in which loop forming dimers are further stabilized by a third C factor.

Conclusions

We presented a chromatin model consisting of a chain carrying reactive monomers folded by bridging (B) molecules dispersed in solution (at a fixed density, ρ_B) reversibly attaching the chromatin fiber. We studied the effect of ρ_B and the association constant between B molecules and the reactive monomers on the morphological properties of the chain. For intermediate values of ρ_B, chromatin folds because of the formation of loops. Instead, overexpression of bridging factors can lead to the unfolding of the fiber. We highlighted the generality of this unfolding transition using thermodynamic considerations. Importantly, the unfolding transition is peculiar to noninteracting bridging factors. Our results can support the analysis of experimental results. In particular, starting from the association constants and the density of the bridging factors, one may infer microscopic interactions between bridging factors starting from the degree of compaction of the fiber.

Our results are supported by Monte Carlo methodologies, which, as compared with molecular dynamics simulations, allow reconfiguring the topology of the networks through dedicated moves, therefore enabling efficient sampling between configurations featuring different connectivity states. Equilibrium configurations feature almost exclusively short loops. Short loops are thermodynamically stable. However, we noticed that it is somehow difficult to relax the system toward the ground state. In particular, longer loops (resulting in much more compact chains) will persist unless using dedicated MC moves implementing multiple reactions in a single step. Our work supports the idea that control of the reaction kinetics (e.g., in the presence of ATP-fueled bridging factors) may play a role in chromatin organization.

Author contributions

B.M.M., I.M., and B.O. designed the research. B.O. and I.M. developed the program. I.M. carried out all simulations and analyzed the data. I.M. and B.M.M. wrote the article.

Editor: Rohit Pappu.