Different relative scalings between transient forces and thermal fluctuations
tune regimes of chromatin organization

Abstract

Within the nucleus, structural maintenance of chromosome protein complexes, namely condensin and cohesin, create an architecture to facilitate the organization and proper function of the genome. Condensin, in addition to performing loop extrusion, creates localized clusters of chromatin in the nucleolus through transient crosslinks. Large-scale simulations revealed three different dynamic behaviors as a function of timescale: slow crosslinking leads to no clusters, fast crosslinking produces rigid slowly changing clusters, while intermediate timescales are optimal for producing flexible clusters that mediate gene interaction. By mathematically analyzing different relative scalings of the two sources of stochasticity, thermal fluctuations and the force induced by the transient crosslinks, we predict these three distinct regimes of cluster behavior. Standard time-averaging that takes the fluctuations of the transient crosslink force to zero can predict the existence of clusters, but not their timescale-dependent lifetimes. Accounting for the interaction of both fluctuations from the crosslinks and thermal noise with an effective energy landscape does capture the timescale-dependent flexible cluster lifetimes. No clusters are predicted when the fluctuations of the transient crosslink force are taken to be large relative to thermal fluctuations. This mathematical perturbation analysis illuminates the importance of accounting for stochasticity in local incoherent transient forces to predict emergent complex biological behavior.

I Introduction

The genome has an intricate and hierarchical organization that allows cells to fit an extraordinary amount of genetic material and perform important nuclear functions within a nucleus mere micrometers in size. Within the nucleus, DNA is complexed with a variety of proteins to form the fibrous material chromatin that makes up chromosomes. Contributing to chromosomal architecture are structural maintenance of chromosome (SMC) protein complexes, namely condensin and cohesin, which facilitate chromosome segregation and gene regulation [18]. Such processes aid in the compaction of genetic material within the nucleus as well as allow the cell to locate and access specific genes depending on cell cycle, cell type, or environmental cue [20]. In particular, condensin helps regulate the retrieval of genetic information by acting on chromatin to form stochastic gene-gene crosslinks [6] and generate loops to induce gene mixing [9, 11, 21, 10].

The emergent behavior induced by condensin crosslinks has been studied computationally with polymer models of chromatin [8, 23, 24, 12, 26, 27] in order to more directly observe the dynamics of DNA over experimental techniques. By representing 5kbp of DNA as beads, linked through worm-like-chain springs to form a long polymer chromatin chain, the timescale of the random SMC crosslink binding between different beads was shown to control the formation of clusters and gene interactions [12, 26]. These clusters, or gene neighborhoods, reveal an underlying organizational framework that optimizes gene interactions, retains a high level of genomic compaction, and yet is highly flexible. A similar detailed level of information is not yet available experimentally; high throughput population methods such as Chromosome Conformation Capture (Hi-C) methods lack temporal evolution information, while microscopic imaging lacks genome-level resolution [22]. The ability to access both the spatial arrangement and the temporal re-arrangement of beads, which is possible in simulation, is crucial for advancing our understanding of life.

Building and remodeling these high-level genetic neighborhoods has the potential to reveal the underlying mechanics for the configuration of the energy landscape within our nuclei. In this paper, we reveal the mathematical mechanism that describes how fluctuations in the model lead to the formation of bead clusters and the temporal mixing of beads between clusters. We vary the timescale of crosslinking forces between beads to change the size of this fluctuating force relative to thermal fluctuations. These various timescales produce three different dynamic regimes. At fast bead crosslinking timescales, rigid, unchanging clusters form. At slow crosslinking timescales, the beads interact in an amorphic state with no clusters. At optimal mean crosslink lifetimes, clusters can both form and exchange beads, indicative of the required gene interaction for proper function. Tuning the timescales of crosslinking provides a unifying mechanism of genome organization that gives insight into how a large number of configurational states can be rapidly remodeled as cells encounter biological challenges.

Configurations that can be rapidly remodeled were shown to take advantage of the interaction between the crosslink binding force fluctuations and the thermal noise fluctuations to promote faster bead exchange between clusters [27]. We build upon this work by incorporating the timescale of the crosslink binding into the mathematical analysis to show how different relative scalings of the two sources of stochastic fluctuations, thermal noise and the force induced by the transient bonds, predict the three distinct regimes of cluster behavior mentioned above. Standard time-averaging that takes the fluctuations of the transient binding force to zero (fast crosslinking timescale) predicts the existence of long-lived rigid clusters. No clusters are predicted when the fluctuations of the transient binding force are taken to be large (slow crosslinking timescale) relative to the thermal fluctuations. Only when accounting for the interaction of both fluctuations from the binding and thermal noise with an effective energy landscape do we recover the flexible clusters and their timescale-dependent mixing.

This work provides the mathematical mechanism underlying the timescale-dependent effects of active agents in biological systems. The timescale controls the production of fluctuations in the forces that the active agents produce. The relative size of these fluctuations to the thermal fluctuations produces different emergent temporal behavior. This mechanism is not restricted to just the chromatin dynamics studied here but is potentially applicable to a wide range of cell-level biological processes such as homology searches for genetic recombination [4, 7, 2], defensive mucus barriers emerging from polymer chains of mucin [28, 17, 25], and error correction through configurational state changes [3, 13, 16]. Thus to deepen our understanding of how such biological systems function, we have shown it is necessary to properly analyze the interplay of different fluctuations and the timescale on which they are generated.

II Results

II.1 Timescale of crosslink binding force drives cluster formation and dynamics

We establish that it is the timescale of the stochastically-switching SMC protein crosslinking force through random binding and unbinding that affects the organization of the beads and the dynamics of the clusters. To show this, we create an idealized 361-bead model that represents the nucleolus, with each bead representing 5 kilobase pairs of DNA in the chromatin chain. The beads bind and unbind based on a biologically feasible random model for the SMC protein crosslinks that is more likely to choose pairs of nearby beads to bind.

Fig. 1 shows the formation of clusters for fast-enough timescale, $\beta$, of the random crosslink binding force. The repulsive excluded volume force and thermal fluctuations acting on the beads are in competition with the attractive stochastically changing pairwise binding forces that coalesce many beads into a cluster. Furthermore, changing the timescale of the crosslink binding force, $\beta>0$, from slow to fast results in three distinct clustering behaviors: The system passes from an amorphic (dissolution of clusters with many bead interactions) to a flexible (frequent bead exchanges between clusters) and then rigid clustering (minimal bead interaction between clusters) behavior.

The observed timescale-dependent clustering behavior mirrors that seen in [26] which employed a larger polymer-like chromosome model and a different random model for the crosslinking proteins in the nucleolus. This demonstrates the universality of the clustering dynamics that are driven by the timescale of the binding forces.

Figure 1: (A-C) Snapshot of 361 bead model for increasing binding timescale. We observe three clustering regimes: (A) amorphic, (B) flexible, and (C) rigid. (D) Mixing coefficient and average nearby number of beads for a range of $\beta$-values for 361 bead model. Flexible clustering occurs for the range of $\beta$-values within the grey region. In this regime, clusters emerge and beads interact.

II.2 Time-averaged force predicts existence of clusters but not their dynamics

To mathematically determine the effective attractive force generated by the stochastic binding we work with a further reduced 3-bead model. We start by showing that a simple time-averaging of the stochastic binding force can predict the existence of clusters but not their timescale-dependent mixing dynamics.

The chosen random model for the crosslinking forces has the added advantage of fitting into a continuous-time Markov chain (CTMC) framework. For the idealized 3-bead model, the four Markov states are easily enumerated: all beads unbound ($s=1$), beads 1 and 2 bound ($s=2$), beads 1 and 3 bound ($s=3$), and beads 2 and 3 bound ($s=4$). The time evolution of the states follows a general CTMC process with a transition rate matrix $S$. Included in $S$ is the bead binding-rate that is proportional to the bead separation distance given by affinity function $a(\cdot)$ and a constant bond breaking-rate $c$.

Now that the binding stochasticity is formulated as a Markov chain, finding the time-averaged force on each bead is straightforward [19]. The percent of time spent in each Markov chain state is given by $\vec{r}(\vec{x})$, the normalized null vector of the switching matrix $S(\vec{x})$, so that the time-averaged force for each coordinate is

$$\frac{dx_{k}}{dt}=\sum_{s=1}^{4}v_{k}(\vec{x};s)r_{s}(\vec{x})=\langle v_{k}(\vec{x})\rangle\;\text{for}\;k=\{1,\dots,6\}$$ (1)

where $v_{k}(\vec{x};s)$ is the force on coordinate $k$ when the system is in Markov state $s$. (Note, we have concatenated the $x$ and $y$ coordinates of each bead into a single vector $\vec{x}$.) This removes the stochasticity and results in a deterministic effective force for the system, $\langle\vec{v}\rangle$.

The stable fixed points of (1) are the observed clusters. One fixed point is a 3-bead cluster in which the beads are in a small triangle configuration; see Fig. 2C. This is distinguished from the unbound state $s=1$, in which the beads form a triangle but with larger pairwise distances; see Fig. 2A. The other three fixed points are 2-bead clusters in which two beads are superimposed and the third is farther away and unbound; see Fig. 2B. These are distinguished from states $s=2,3,4$ as the clusters persist longer than any single bond lifetime.

In addition to predicting the existence of clusters, we can test this time-averaging procedure’s ability to predict the lifetime of a cluster. We do this by considering the effects of small perturbations of noise about the average by studying the equation,

$$dx_{k}=\langle v_{k}(\vec{x})\rangle dt+\sqrt{2\epsilon}\;dB_{k}$$ (2)

in the limit as $\epsilon\to 0$. Even though the vector of forces is the gradient of a potential function for each state $s$, i.e. $\vec{v}(\vec{x};s)=-\nabla_{x}U(\vec{x};s)$, it is not true that $\langle\vec{v}(\vec{x})\rangle=-\nabla_{x}\langle U(\vec{x})\rangle$ since the CTMC null vector $\vec{r}$ depends on the positions $\vec{x}$. However, we numerically find an effective potential $U_{\text{eff}}$ such that $-\nabla_{x}U_{\text{eff}}(\vec{x})\approx\langle\vec{v}(\vec{x})\rangle$ along the most probable transition path connecting a minimum of $U_{\text{eff}}$ (recall these are the stable fixed points of (1)) to a saddle point using the String Method [5]. We are therefore justified in asymptotically approximating the escape times from a cluster using the well-known Arrhenius law,

$$\log(\mathbb{E}[\tau])\sim\frac{\Delta U_{\text{eff}}}{\epsilon}$$ (3)

in the limit as $\epsilon\to 0$, where $\Delta U_{\text{eff}}$ is the change in the effective potential generated by the time-averaged force along the most probable transition path [29].

Figure 2: (A-C) plots of pairwise distances of three beads over time and a snapshot of the three bead configuration: (A) all beads unbound: balance between excluded volume and confinement forces keep beads in a large triangle configuration, (B) 2-bead cluster state with beads 1-2 bound, (C) 3-bead cluster state: all beads remain close, rapid switching between pairwise bonds keeps beads in a small triangle configuration.

Despite predicting the existence of clusters, the naive time-averaged force does not produce an effective potential $U_{\text{eff}}$ that accounts for the observed differences in the cluster lifetime as the crosslink binding timescale $\beta$ varies in the 361-bead model. The time-averaged force, and thus its effective potential, is invariant to the overall timescale of the transition rate matrix. Scaling $S$ by some constant $\alpha$ does not affect the fraction of time spent in each state because $\vec{r}$ is also in the null space of $\alpha S$. Thus, Eqs. (1) and (2) are unchanged when scaling the transition rate matrix $S$ by $\alpha$ and therefore cannot predict subsequent changes to the dynamic cluster lifetimes.

II.3 Relative scalings of fluctuations predicts three different clustering regimes

The key to understanding clustering dynamics is accounting for fluctuations from both the stochastic binding and the small perturbations of thermal noise. To analyze the dynamics mathematically we perform an asymptotic expansion as the size of the fluctuations goes to zero. We consider taking both fluctuations to zero at the same rate and at different rates. The latter allows one noise source to dominate. Rigid clusters arise when the binding fluctuations are small and only thermal noise drives transitions between clusters. Amorphic arrangements arise when the binding fluctuations are large and take place on a longer timescale than the fast thermal fluctuations. This binding drives transitions between clusters. Taking fluctuations to zero at the same rate allows for interaction between the noises. We find that this interaction is required to predict the flexible cluster dynamics that depend on the timescale of the binding. Together, these three asymptotic regimes explain the three regimes seen in [26] and Fig. 1.

We include the effects of fluctuations about the mean force so that the position of each bead follows the overdamped Langevin-like equation given by

$$dx_{k}=v_{k}(\vec{x};s)dt+\sqrt{2\epsilon}dB_{k}.$$ (4)

where $s$ is the state of the CTMC. How we choose to scale the transition rate matrix $S$ determines how the relative rates of the two different fluctuations go to zero.

We start with the scaling $\frac{1}{\epsilon^{2}}S$ which takes the binding fluctuations to zero faster than the perturbations from thermal noise. We seek an effective potential $W$ to replace $U_{\text{eff}}$ in Eq. (3). Therefore we assume a steady-state distribution

$$p_{s}(\vec{x})=r_{s}(\vec{x})\exp\bigg{(}-\frac{1}{\epsilon}W(\vec{x})\bigg{)}$$ (5)

as the asymptotic solution for the system of steady-state Fokker-Planck equations,

$$\frac{\partial p_{s}}{\partial t}=-\sum_{k=1}^{6}\frac{\partial}{\partial x_{k}}[v_{k}(\vec{x};s)p_{s}]+\epsilon\sum_{k=1}^{6}\frac{\partial^{2}}{\partial x_{k}^{2}}[p_{s}]+\frac{\alpha}{\epsilon^{2}}\sum_{j=1}^{4}S_{sj}p_{j}.$$ (6)

The element of the CTMC transition matrix $S_{sj}$ gives the transition rates from state $j$ into state $s$. Thus, the last term couples the process between different states. Setting (6) to zero and plugging in (5) yields the leading order equation

$$S\vec{r}=0$$ (7)

with next order equation

$$r_{s}\vec{v}\cdot\nabla_{x}W+r_{s}\nabla_{x}W\cdot\nabla_{x}W=0.$$ (8)

Thus, $\vec{r}$ is the steady-state distribution of $S$ and summing Eq. (8) over $s$ reveals that $-\nabla_{x}W=\sum_{s}\vec{v}(\vec{x};s)r_{s}(\vec{x})$ is exactly the naive time-average from above that does not depend on $\alpha$.

If instead we take the scaling $\frac{1}{\epsilon}S$, both sources of noise show up in the leading order equation given by

$$[A(\vec{x},\nabla_{x}W)+D(\nabla_{x}W)+\alpha S(\vec{x})]\vec{r}=0$$ (9)

for advection (drift) matrix $A$, diffusion matrix $D$, and switching rate matrix $S$; see [27] for details. In this regime, both sources of noise are important for flexible clustering. We see the effects of varying the binding timescale parameter $\alpha$, which controls how quickly the simulation switches between the Markov chain states. This role mirrors that of the kinetic timescale parameter $\mu$ in the large-scale simulations of [12, 26] in which $\mu$ is a “tuning knob” used to set the kinetic timescale on which the crosslinks bind and unbind.

To further illustrate the changing dynamics with $\alpha$, we compare the predicted lifetime of the cluster governed by Eq. (3) with Monte-Carlo simulations. The asymptotic escape times from one 2-bead cluster to another 2-bead cluster are shown in Fig. 3. The slope value is given by the quasipotential barrier height and well expresses the linear relationship of the mean escape times. As $\alpha$ increases, the effective energy barrier approaches the barrier predicted by the naive time average discussed previously.

Figure 3: Transition: 2-bead cluster to 2-bead cluster. (a) Quasipotential (solid) for different $\alpha$ values and for the time-averaged binding force (dotted) along the transition path, normalized to unit length. (b) Comparison of asymptotic escape times computed via Monte Carlo simulation to slope taken from quasipotential barrier height. (c) Quasipotential energy barrier height vs. switching timescale $\alpha$.

Finally, if we keep $S$ independent of $\epsilon$, retaining all the fluctuations, the system equilibrates within each Markov chain state and there are no clusters predicted. In this regime, the effects of the binding noise are pushed into a higher order so that the leading order does not contain any contributions from the CTMC. We observe the beads are often unbound, settling into a “large triangle” configuration; see Fig. 2A. This long-observed state of the system is a balance between the excluded volume and the confinement forces. We do not consider this a cluster as it would dissolve in the absence of the confinement forces. Fig. 4 shows the formation of clusters as $\alpha$ increases. The system is able to remain in a cluster state despite all the beads being unbound as the system does not have time between binding events to reach the large triangle equilibrium.

Figure 4: Average proportion of simulation spent in $s=1$ unbound state of the Markov chain and average proportion of simulation not spent in a cluster state vs. $\alpha$.

II.4 Effects of crosslink binding fluctuations on different transitions

Scaling $S$ by $\frac{1}{\epsilon}$ revealed the dependence of cluster lifetimes on the crosslink binding timescale, $\alpha$. By accounting for both sources of randomness in the system we computed an accurate quasipotential that predicted a higher energy barrier height between minimum states as $\alpha$ increased, and thus longer cluster lifetimes. However, this increase is not uniform across all transitions as we show in Fig. 5.

Of the ‘uphill’ most probable transition paths the system takes from one minimum state to the saddle point shown in Fig. 5A-D, only the 3-bead on its way to a 2-bead cluster has noticeable dependence on $\alpha$ (Fig. 5C). This transition also has the largest change in quasipotential energy barrier with $\alpha$ (Fig. 5F) indicating removing a bead from a cluster is more sensitive to the binding timescale that rearrangements within the cluster (pathway in Fig. 5D). The transition shown in Fig. 5D to the “collinear” saddle point configuration also has the lowest energy barrier height amongst the transitions, with limited dependence on $\alpha$ for either the pathway or the height. This indicates that the binding force is not important for this transition, as the stochasticity of the binding does not help the system make the transition.

Starting in the 2-bead cluster state, our system can transition into a different 2-bead cluster (Fig. 5A) or into the “small triangle” 3-bead cluster (Fig. 5B). These transitions have similar pathways up to each saddle point that do not significantly depend on $\alpha$, and have similar quasipotential energy barrier heights for the range of $\alpha$ values from 0 to 20. This suggests that the system will take either transition with similar probability and do so in approximately the same amount of time.

Figure 5: (A-D) Transition pathways to a saddle point: (A) 2-bead cluster on the way to a different 2-bead cluster (B) 2-bead cluster on the way to a 3-bead cluster, (C) 3-bead cluster on the way to 2-bead cluster for $\alpha=1$, $\alpha=10$, and time-averaged binding force ($\alpha\to\infty$), (D) 3-bead cluster to the collinear saddle point configuration that continues to a rearrangement of the 3-bead cluster, or to a different saddle point on the way to a 2-bead cluster. (E,F) Energy Barrier heights vs. $\alpha$ for these transition pathways

III Discussion

The underlying mathematical mechanism behind the observed chromatin clustering behaviors is the competition between thermal fluctuations and the timescale of the model condensin crosslinking force. This timescale controls the production of random fluctuations in the crosslink binding force and in turn the dominating terms in the asymptotic perturbation when searching for an effective thermal equilibrium. This analysis emphasizes the importance of accounting for stochasticity in local incoherent transient forces to predict emergent complex biological behavior.

The key to understanding flexible clustering dynamics is accounting for fluctuations from both the stochastic binding force and the thermal noise, taking both fluctuations to zero at the same rate. To produce this interaction mathematically, we scale the binding rate matrix $S$ by $\frac{1}{\epsilon}$. Taking $\epsilon\to 0$ with this scaling, an effective binding force remains that allows for stable clusters. This limit also accounts for the fact that the fluctuations of the binding force aid in transitions between clusters, as it creates finite periods of time when the thermal noise has to overcome smaller forces to push the system into a new cluster state. Since our added binding timescale parameter $\alpha$ controls the size of the fluctuations, it too controls the size of the effective energy barrier for predicting transition times between stable cluster states given by the Arrhenius law.

If we take the fluctuations to zero at different rates, one source of noise “dominates” over the other in the limit as $\epsilon\to 0$. Choosing to scale $S$ by $\frac{1}{\epsilon^{2}}$, the fluctuations of the binding force go to zero faster than the thermal fluctuations. An effective binding force remains that allows for stable clusters, but the fluctuations in the binding force are now much smaller and do not interact with the thermal noise to aid in transitions between clusters. Indeed, we recover the naive time average to describe the effective energy barrier, which is also the limiting barrier as $\alpha\to\infty$ of the above-mentioned fluctuation-interaction case.

If we flip which fluctuation source dominates in the limit as $\epsilon\to 0$ by keeping $S$ independent of $\epsilon$, switching between Markov chain states occurs so infrequently that the system equilibriates while in this one state. No effective force is generated by the superposition of different states, thus no clusters are predicted. In the 3-bead model, two cluster-like states appear but their generating mechanism would not produce clusters in the 361-bead model. The first is a 3-bead triangle configuration, but it is a result of the balance between the repulsive excluded volume force and confinement force. In the 361-bead model, this would correspond to all beads in an amorphic arrangement. The second is a 2-bead-like cluster formed by the fact that a crosslink binds two beads; this cluster returns to the 3-bead triangle configuration once the bond is broken. In the 361-bead model, this would correspond to another amorphic arrangement with many pair-wise bound beads.

The arrangement of beads within clusters provides a physical model for the dynamics of gene clusters within the nucleus. Gene clustering is a mechanism that facilitates the co-regulation of un-linked genes or of long linear arrays of repeated genes such as rDNA. The ability to reconfigure the network is central to the cell’s ability to re-wire its transcriptional circuitry. Using a relatively small number of beads to reveal root mechanisms, we account for both state changes and dynamics between states with thermal fluctuations and cross-linking forces. The energy barrier for rearrangements within a cluster is lower than that required for exchanges between clusters (Fig. 5E, F). The system is remarkably plastic, with small bursts of energy able to potentiate new configurations depending on the biological demand.

The emergence of a wide range of clustering regimes (amorphic, flexible, rigid)(Fig. 1) and exchange within and between clusters with a pared-down 3-bead model (Fig. 5) provides a framework for understanding the governing principles of genome organization: The continuous chain of each chromosome biases their individualization. The crosslinking forces provide a mechanism to build gene (bead) clusters and generate informational circuits that are agnostic to the position of a gene in a given chromosome. This work shows that by controlling the timescale of the crosslinking forces, the plasticity of the gene clusters can be tuned based on biological needs. The predictions of cluster dynamics for larger systems will depend greatly on thermodynamics and tuning the crosslinking forces. This model and analysis provide critical insight into the diversity of network configurations with a minimal set of parameters.

Computationally, our analysis is limited to relatively small numbers of beads due to the need to enumerate all possible pairwise bound states for the Markov chain. With the growing power of machine learning, it is likely possible to build a physics-informed neural network [14, 15] to learn the effective potential. This would allow predictions of cluster dynamics for larger systems but would need to be re-applied for each timescale or set of model parameters. The effective potential alone gives little insight into the mechanism underlying the emergent behavior that our analysis of the 3-bead model has provided. This new outlook on the importance of noise at the proper timescale aids in deepening our understanding of life at the cellular level.

IV Methods

The idealized chromatin model is based on the polymer-like chain of beads model in [12, 26]. The position of bead $i$ is given by the overdamped Langevin equation

$$d\mathbf{x}_{i}=(\mathbf{f}_{c}^{i}+\mathbf{f}_{\text{EV}}^{i}+\mathbf{f}_{\text{bond}}^{i})dt+\sqrt{2\epsilon}d\mathbf{B}.$$ (10)

The parameter $\epsilon$ scales the vector of Brownian increments $d\mathbf{B}$ preparing for asymptotic analysis to take the fluctuations to zero. The deterministic confinement force,

$$\mathbf{f}_{c}^{i}=-\eta\mathbf{x}_{i},$$

replaces the hard-wall constraint of the nucleus membrane while the excluded volume force,

$$\mathbf{f}_{\text{EV}}^{i}=\sum_{j\neq i}a_{ev}(\mathbf{x}_{i}-\mathbf{x}_{j})\exp\bigg{(}-\frac{|\mathbf{x}_{i}-\mathbf{x}_{j}|^{2}}{c_{ev}}\bigg{)},$$

remains similar to its form in [12, 26]. Parameters $\eta$, $a_{ev}$, and $c_{ev}$ are given in Table 1 for both the 361-bead $(\mathbf{x}_{i}\in\mathbb{R}^{3},i=1\dots 361)$ and the 3-bead $(\mathbf{x}_{i}\in\mathbb{R}^{2},i=1\dots 3)$ models. Note we have neglected the spring force linking the beads together to form a chain, as we show it is secondary to the clustering dynamics.

The stochastic binding force,

$$\mathbf{f}_{\text{bond}}^{i}=\sum_{j\neq i}\kappa b_{ij}(\mathbf{x}_{j}-\mathbf{x}_{i}),$$

models the binding SMC proteins found in the biological system. This force binds two beads, corresponding to $b_{ij}=1$ if beads $i$ and $j$ are bound and 0 otherwise. Each bead can be bound to only one other bead; these stochastic bonds form and break at exponentially distributed times with a binding-rate proportional to the bead separation distance, $a(\mathbf{x}_{i}-\mathbf{x}_{j})$, and constant breaking-rate $c$. The time evolution of the states follows a general CTMC process; for the 3-bead model the transition rate matrix is given by

$$S=\begin{pmatrix}b&c&c&c\\ a(\mathbf{x}_{1}-\mathbf{x}_{2})&-c&0&0\\ a(\mathbf{x}_{1}-\mathbf{x}_{3})&0&-c&0\\ a(\mathbf{x}_{2}-\mathbf{x}_{3})&0&0&-c\end{pmatrix}$$ (11)

with $b=-a(\mathbf{x}_{1}-\mathbf{x}_{2})-a(\mathbf{x}_{1}-\mathbf{x}_{3})-a(\mathbf{x}_{2}-\mathbf{x}_{3})$.

The affinity function $a(\mathbf{x})$ and parameter values are given in Table 1 for both the 361-bead and the 3-bead models. Both $a(\mathbf{x})$ and $c$ are scaled by $\beta$ for the 361-bead model and $\alpha$ for the 3-bead model to explore the kinetic timescale on which the crosslinks bind and unbind, mirroring the kinetic timescale parameter $\mu$ in the large-scale simulations of [26]. Both $a(\mathbf{x})$ and $c$ are further scaled by different powers of $\epsilon$ to perform the asymptotic analysis of the fluctuations.

Table 1: Model parameters and formula.

	361-Bead Model	3-Bead Model
$a(\mathbf{x})$	$\displaystyle\frac{2}{1+\exp({20(|\mathbf{x}|-75)})}$	$\displaystyle\frac{2}{1+\exp({20(|\mathbf{x}|-0.75))}}$
$\eta$	0.002	1
$a_{ev}$	0.0332	2
$c_{ev}$	30600	0.5
$\kappa$	10.9	5
$c$	0.01	0.5

Monte Carlo Simulations

We perform Monte Carlo simulations of the 3-bead system to compare the scaling of the escape times with $\epsilon$ to the effective energy barriers predicted by the theoretical calculations. The simulations are started in either the 2-bead or 3-bead cluster, and continued until a stopping condition is met, indicating that the system has left the basin of attraction of the initial state. From the 2-bead cluster, we look for either a different 2-bead cluster or the 3-bead triangle cluster. From the 3-bead triangle cluster, we look for either a 2-bead cluster or a collinear configuration that is the saddle point between different arrangements of the 3-bead triangle cluster. The mean escape time, $\tau$, is computed by the maximum likelihood estimate that divides the sum of all escape times by the number executing the desired transition.

Code for reproducing results is available on GitHub [1].