Genomic Energy Landscapes

Abstract

Energy landscape theory, developed in the context of protein folding, provides, to our knowledge, a new perspective on chromosome architecture. We review what has been learned concerning the topology and structure of both the interphase and mitotic chromosomes from effective energy landscapes constructed using Hi-C data. Energy landscape thinking raises new questions about the nonequilibrium dynamics of the chromosome and gene regulation.

Main Text

The genome is an enormously large, information-rich molecular structure. More than 10,000,000,000 bits of data are encoded, each roughly a nanometer in size, in a meter of polymer that can take up less than a cubic micron in the nucleus. These data must be accessed and manipulated by the cell’s active machinery to respond to environmental signals and be copied as cells divide. How are the data organized in space and time? What are the forces needed to accomplish this? Energy landscape theory holds promise as a framework for crisply asking and answering these questions.

Energy landscape theory has proved useful for understanding the folding and function of molecules that are much smaller than genomes, proteins (1). Proteins generally exist both in well-organized folded states and disordered unfolded ensembles of configurations. These ensembles interconvert, both globally and locally, while remaining nearly at thermodynamic equilibrium. The functional landscape of proteins overlaps the folding landscape that organizes them (2). Energy landscape theory characterizes the folding of proteins using ideas borrowed from the theory of phase transitions and glass transitions, but applies these ideas in the context of a system of finite size. In this way the theory provides a statistical description of the free energy basins and the kinetics of transitions between basins. Energy landscape theory has been used to characterize the ordered and disordered ensembles of proteins (3), and to provide criteria for how foldability has evolved by selecting for a “funneled landscape” (4). Models based on energy landscape ideas have been successfully used to describe the mechanisms of folding and function in quantitative detail (5, 6). The theory also provides mathematical tools for learning the forces that fold proteins using known structural data (7).

Can energy landscape ideas be used to study the spatial structure and dynamics of the entire genome? It is not obvious that they can. There are several potential difficulties. In practical terms, our current experimental knowledge of genome spatial organization is poor, in comparison to the knowledge of protein structure available from x-ray crystallography. Partly this lack of knowledge comes from the size scales of the genome falling in the no man’s land between x-rays and visible light (8). Also, in contrast to the situation for many proteins, the genome can never globally be referenced to just a single structure (9). The single structure assumption, while not perfect, greatly simplified the interpretation of protein experiments. The ensemble language of statistical mechanics, the cornerstone of energy landscape theory, is therefore needed from the get-go for chromosomes. Perhaps more problematic, many aspects of the dynamics of genomes seem to be kinetically controlled, as would be expected for physically large objects. Does this nonequilibrium aspect destroy any possibility of using energy landscape theory, which, while dynamical, employs many ideas from equilibrium statistical mechanics?

While acknowledging these difficulties, in several recent articles (10, 11, 12), we have undertaken to see how far energy landscape theory can be used to study chromosomes. As was done for proteins, we start from experimental structural data, coarse grained as it is presently for genomes. We have used Hi-C data to construct effective energy landscapes for chromosomes that reproduce the detailed information provided by Hi-C about the contacts between different genomic segments. A cross-linking experiment, Hi-C, resembles the fluorescence resonance energy transfer and NMR nuclear Overhauser effect techniques used in the protein world to measure physical contacts. Hi-C data are typically collected over a population of cells and must contend therefore with a mixture of configurations, although a single-cell Hi-C protocol has also recently become available (13). The effective energy landscape inferred from these data nevertheless leads to an interesting and detailed structural picture of how the ensemble of chromosome structures is organized both for the cell in interphase states, where the cell spends most of its life and some genes are active (10), and during the metaphase, where the genome is largely quiescent and the chromosomes are condensed in preparation for cell division (11). The interphase chromosome turns out to be globally disordered, but has fluctuating order on moderate length scales (10). The locally ordered regions can act as cooperative switches, opening and closing, much like individual folding proteins. In contrast, the mitotic chromosome turns out to be globally ordered and possesses obvious macroscopic structure. The study of these effective energy landscapes determined directly from experimental data also has suggested the form for an effective force field that is transferable, i.e., that can predict detailed Hi-C data for many distinct human chromosomes, starting only from knowledge available from binding assays about the epigenetic state of contiguous regions on each chromosome (12).

In this perspective, we first briefly review the energy landscape methodology developed for interpreting Hi-C data. We then review predictions from the resulting landscapes about the topology, local order, and global structure of the genome. We then explore several questions raised by energy landscape analysis, including: how is molecular level information transferred from the DNA sequence level to the global genome structure? Because the chromosome is not fully at equilibrium, is the effective energy landscape inferred from the Hi-C experiment quantitatively correlated to the actual physical forces sculpting the genome? Do history dependent effects dominate? Can we understand the timescales of the fluctuating genome? Do these timescales modify the stochastic dynamics of gene regulation and epigenetics?

Learning chromosome landscapes

To understand structure, a powerful approach is to start from the bottom and move up, i.e., to construct an energy landscape for the chromosome based on the physicochemical interactions of all its components. A start on this has already been made for the nucleosome (14, 15, 16, 17, 18) and for nucleosome oligomers (19, 20, 21, 22, 23) where all the components are known. Many of the proteins in the full chromosome remain unknown however, so scaling up this direct physicochemical strategy to the entire cell nucleus takes some patience and some courage.

It therefore seems sensible also to embark upon a top-down approach employing directly experimental data on large length scales. Because the chromosome is so large, it would appear that the probability distribution of its structures must practically factorize over the many parts of the chromosome, i.e., the conformational distribution can be fit by an exponential of an additive effective energy function. The quasi-independence of parts of large systems is the main idea behind the maximum entropy approach that also has been used in many areas of science where the fundamental components are poorly known (24). Assuming that the set of chromosome structures from a population of cells can be described by a quasi-Boltzmann distribution, we can use inverse statistical mechanics to construct an effective energy landscape from Hi-C data on large length scales. Hi-C, a genome-wide version of the chromosome conformation capture method, measures the frequency at which any given pair of genomic segments comes into close contact inside the nucleus (25, 26). If we acknowledge that the chromosome is a connected chain molecule but that its effective Boltzmann distribution must also reproduce the Hi-C data, then the maximum entropy principle implies the effective landscape in the quasi-Boltzmann distribution takes the form:

$$U_{\text{ME}}\left(r\right)=U\left(r\right)+\sum _{i,j}{\alpha }_{i,j}f\left(r_{ij}\right),$$

where $U\left(r\right)$ is the potential energy function of a homopolymer with bonds and excluded volume to maintain the chain integrity. The expression $f\left(r_{ij}\right)$ is a function giving the probability that a contact will be recorded when a pair of genomic segments i and j is separated by a spatial distance $r_{ij}$. The coefficients $\left\{{\alpha }_{i,j}\right\}$ describe the strengths of individual contact potentials between different parts of the genome, and can be derived via an iterative approach using the experimental Hi-C data along with the averages and standard deviations of the contact counts found from simulations with the effective landscape. Iteration is carried out until the simulated and experimental contact probabilities agree. This procedure for learning an effective potential from structural data mathematically resembles the determination of statistical potentials from databases of protein structures using Z-score optimization (27), an approach derived from energy landscape theory.

Exploring data-based chromosome landscapes

A typical interphase configuration sampled from simulating the interphase landscape is shown in Fig. 1 A (10). Overall, the chromosome at interphase is isotropic and roughly spherical. Regions with differing levels of high gene activity (indicated by the color red) cluster together and appear to phase-separate from low activity regions (in white), showing the coupling of chromosome structure and gene activity.

Figure 1.

Example structures of the interphase (A), the quenched interphase (B), the ideal (C), and the mitotic chromosome (D) predicted by the energy landscapes inferred from Hi-C data (33, 74). In (A), the structure is colored by gene expression determined from RNA-Seq experiments (75), with red and white representing high and low expression, respectively. In (B–D), the structure is colored by genomic distance. To see this figure in color, go online.

Once an energy landscape is inferred from the experimental data, statistically sampling configurations allows us to look at nonlocal aspects of chromosome structure and dynamics, features that are not directly visible in the Hi-C data themselves. Topology is an example of such a nonlocal feature. The topological problems of handling long genomes were evident already to Watson and Crick (28), who in their second publication on DNA, wondered how the double helix could be duplicated without becoming entangled. Equilibrated long polymers when collapsed should have knots. Knots being global constraints take enormous times to untangle spontaneously (29). Topoisomerases, enzymes that change global DNA topology by local motions, essentially cut the Gordian knot. They are needed for replication and are present at high concentration in the cell nucleus (30). Sampling the chromosome landscape allows the question of the topology of the chromosome to be answered directly by computing various knot invariants for sampled configurations. The most intuitive method to employ is to compute the average minimal rope length (31), which is found by joining the ends of the DNA and computationally shrinking the chain along its length but not its girth, never allowing the chain to pass through itself. A completely unknotted chain shrinks to a point while a very complex knot will have a large minimal length. Using this method we found the interphase chromosome structures are mostly unknotted, having at most a trefoil knot, while an equilibrated chain with a trivial energy landscape retaining only connectivity and weak excluded volume contains quite complex knots. The simulations also show that with sufficient amounts of topoisomerase these relatively knot free ensembles can form in timescales less than the hours-long cell cycle, but would be unable to form if there were no topoisomerases.

All the analyses on chromosome topology were performed over an essentially equilibrated ensemble of configurations that is consistent with the inferred energy landscape. Equilibrium is achieved because the polymer’s soft core repulsion potential in the simulations is limited to a finite value (4 k_BT) at zero distance. This allows chain crossing in the computer simulations and expedites the topological relaxation of polymer configurations while nevertheless providing an accurate description of the thermodynamic excluded volume effects. Biologically, this finite repulsive core potential mimics the effect of topoisomerases, which allow chains to pass through each other, although at a crude level. The conclusions on chromosome topology from this model are therefore largely independent of simulation details, such as the initial configurations.

In the effective energy landscape, knots are prevented from forming by virtue of the chain being locally more rigid than unstructured DNA. This rigidity comes from forming local structure. While the interphase chromosome is globally disordered, some local structures of the chromosome are favored over others. Energy landscape theory suggests a way to find the favored structures. Using the information theoretic energy function, configurations are quenched to a much lower effective information theoretic temperature than that of the self-consistent statistical sampling. Such structures, representing energy minima, are known as “inherent structures” in liquid theory (32). For a protein, quenching denatured structures yields the native configuration because the landscape is globally funneled. A quenched structure of the interphase chromosome is shown in Fig. 1 B. At length scales <1 megabase, the statistically sampled chromosome configurations greatly resemble their fully quenched counterparts but on larger length scales, this order is lost through fluctuations. On the megabase length scale, some regions display a two-state-like free energy profile, with one subensemble matching the quenched local structure very well, while the other subensemble is disordered. These regions, called “topologically associated domains” (TADs) (33, 34), thus can be said to fold and unfold. The local free energy landscape of TADs varies heterogeneously along the genome. Some TAD free energy profiles are more clearly bistable than others are. The activated transitions between folded (closed) and unfolded (open) TADs can modulate the binding of transcription factors to the gene to regulate it (35, 36, 37, 38). The structures of the locally ordered domains seem complex. Some of the apparent structural complexity of the folded TADs doubtless reflects true biological heterogeneity but some variations may also come from unavoidable error in the input Hi-C data used to create the landscape. To find a simple description of the order, we signal-average by constructing an ideal chromosome model landscape that averages over different regions to create a landscape for a uniformly self-interacting chromosome that is sequence-translation invariant. The quenched structures of the ideal chromosome energy landscape resemble those formed from the full landscape locally, but on the global scale comprise a beautifully regular hierarchically layered fiber of fibers shown in Fig. 1 C. The quenched ideal chromosome is a crystalline cylinder that on large length scales resembles the familiar metaphase chromosome as visualized in the light microscope (39).

We can also use the maximum entropy strategy directly to invert Hi-C data obtained for the metaphase chromosome during mitosis. This landscape even at high temperatures leads to a structured ensemble that indeed does resemble the quenched ideal chromosome but that fluctuates on shorter length scales (see Fig. 1 D). The mitotic chromosome is not a crystal, but is clearly an anisotropic liquid crystal that breaks rotational symmetry to form a cylinder. Large segments of the mitotic chromosome also break parity symmetry being either right-handed or left-handed spirals. The global breaking of parity symmetry predicted by this landscape has been seen in microscopic studies of sister chromatids, which are mirror image pairs directly after the genome is duplicated (40).

Several ingredients that emerge from the direct-data-driven energy landscapes suggest a form of a transferable force field that is able to predict chromosome structure using independent laboratory information as input (12). The ideal chromosome landscape is one essential ingredient of the predictive model. This homogeneous landscape captures the generic tendency of the chromosomes to order locally like liquid crystals. Another essential ingredient gleaned from the full inversions used in the transferable model is the tendency of regions with different levels of gene activity to phase-separate. Examining the contact maps themselves already suggested that chromatin regions can be classified into five types, reflecting different tendencies to bind different proteins and thus to be active in different ways (41). Finally, an important feature again directly visible in the Hi-C data, the limits of individual TADs can be predicted by local signals in the DNA sequence that bind the CTCF protein as a cross link (42). An energy landscape containing these three features by themselves is really a special case of the landscape used for full inversion, but because the transferable model has many fewer parameters, the optimal parameters can be found using the data from one single chromosome, chromosome 10. Once these parameters are found by iteration, again only for this particular chromosome, the parameters can be transferred to predict the structural ensembles for other chromosomes. The transferable landscape not only gives similar quality agreement with the Hi-C data as found using the full inversion for the input chromosome, but happily, also predicts the Hi-C data for all other human chromosomes using knowledge of their chromatin types along the sequence!

Pressing open questions

The origins of order

The ensembles of structures found by using the energy landscape maximum entropy algorithm to directly invert the Hi-C data reveal ordering on scales dramatically larger than the nucleosome (33, 43, 44, 45). The chromosome, even in the less dense interphase state, is far from being an ideal polymer of nucleosomal thread. There are >200 nucleosomes in even a single bead of our landscape simulations. Nevertheless, several kinds of order are revealed at the supernucleosomal scale: local formation of domains (i.e., TADs), liquid crystalline ordering both within and across these domains, and spatial segregation of domains having different chromatin types. To find the origins of this order, filling in the scale gap between the nucleosome and each bead is a pressing need.

How are the topologically associated domains formed?

A key element of forming TADs is clearly the formation of the specific cross links determining them (46), though other effects could also play a role (47, 48). Like the specific covalent disulfide bonds in proteins, which contrast with weaker hydrophobic interactions throughout the protein, these links turn out to be much more stable than the other interactions along the genome chain. Are cross links equilibrium structures? It has been argued that they cannot be formed by equilibrated cross linking by a diffusible protein component but instead are made by a kinetically controlled extrusion complex—much as disulfides are frozen in by quick oxidation (42, 49, 50). The rationale for a nonequilibrium cross linking is that knowledge of the sequence orientation of the DNA (in the 5′–3′ direction) is preserved to a very large extent when the cross links are formed. This argument is quite strong, but in our opinion, not entirely ironclad. The rigidity of the chromosome on the very longest length scales suggests that rigidity of the double helix itself cannot be ruled out on the TAD scale. Supercoiling of the DNA can lead to the formation of plectonomes, objects familiar to many of us for messy telephone cords in the pre-cell phone era. Plectonemes could preserve orientation memory on the TAD length scale (51, 52). There have also been reports of deviations from the strict sequence orientation rule (53).

What leads to the local liquid crystalline ordering?

DNA itself, when dense enough, forms liquid crystals. Decorating the DNA with proteins, as in the chromosome, increases both its local girth and rigidity, so liquid crystal ordering of the super helix would be encouraged. Is this effect sufficient? So far it is not clear: nonequilibrium mechanisms may also play a role.

What causes the observed large-scale spatial segregation?

It is important to bear in mind that polymers of very high molecular weight naturally do not mix: even protonated and deuterated versions of high polymers of sufficient molecular weight phase-separate (54)! Very small differences in the forces between monomer units are able to overcome the diminished entropy of mixing of a long connected polymer, according to Flory’s theory (55). Completely random DNA sequences can mix but if the variations in composition in large regions are consistent rather than random, those regions will segregate (56). Are the consistent differences that are uncovered by binding assays of chromatin type alone (57), enough at equilibrium to give rise to the observed phase segregation? Possibly, the phase segregation seen in active matter (58) where active particles phase-separate from passive ones also plays a role.

How important is active dynamics in general?

In considering the energy landscape analyses of the chromosome, we must confront questions not just of structure but also of kinetics. Fig. 2 shows a range of timescales that can be assigned to structural changes on different length scales. As we have noted already, large objects are seldom under purely thermodynamic control. Kinetic barriers increase with system sizes; for protein folding the barriers increase with length N as N^2/3 roughly (59). If barriers increase polynomially with length, reaction times would increase exponentially, unless special mechanisms come into play. There is ample room for such mechanisms in the cell nucleus: the chromosome is motorized! The chromosome is tugged at by numerous actors in the cell: polymerases, topoisomerases, remodeling proteins as well as the cytoskeleton. Do these active processes vitiate an energy landscape description? Not necessarily! Models of motorized crystals and glasses, proposed in the context of cytoskeleton models, show that if the step sizes of the induced motions are smaller than the characteristic length scales of the forces, an active motorized system can still be described as a quasi-equilibrium system but with renormalized interactions (60) and an effectively high temperature (61). If the steps sizes are larger than the range of the forces, however, sustained flows can develop that require going beyond a landscape description (62). Answering which limit is most relevant for chromosomes requires comparing landscape simulations with dynamic observations. Certainly the pulling apart of daughter chromosomes during mitosis requires sustained vectorial motions but how important directed motions are, at the length scales described in this energy landscape analysis of the chromosome, remains to be seen. If the chromosome is able to come to an effective albeit motorized equilibrium, the forces produced by the effective landscape will correlate with real mechanical forces that can be externally applied. Carrying out simulations of pulling experiments and comparing them quantitatively to laboratory data (63) could quantify the effective motorized temperature.

Figure 2.

Illustration of the timescales associated with various structural transitions on different length scales that can occur in the chromosome. To see this figure in color, go online.

Can the chromosome actually be effectively equilibrated by the motors or does it still remember how it was formed (in contrast to the landscape simulations)? One of the most theoretically attractive ideas about chromosome structure has been the notion that the chromosome is a “fractal globule” (26, 64, 65). When a very long chain collapses from its typically unknotted low density state with sufficient rapidity, knots do not have time to form and thus pair contact probabilities will differ from those of a fully equilibrated chain (66). This phenomenon can be captured by an effective landscape: the nonknotting constraint would introduce an effective pair interaction. While the idea of history dependence requires further investigation, the original purely topological argument is less than self-evident because of topoisomerases. Current thinking also inclines against the argument, because not all chromosome Hi-C data conform to the fractal globule’s universal predictions (42, 67). Nevertheless, some remnant of the topological jamming may play an important part in the effective landscapes.

History may be important in another way. Simulations of the effective energy landscapes already show a broad range of timescales in chromosome dynamics (see the Supporting Material of Zhang and Wolynes (10)). This sort of glassy temporal heterogeneity has also been seen in the laboratory, where markers along the chromosome exhibit anomalous diffusion. Perhaps the most interesting dynamics to study occurs on medium length scales, the folding and unfolding of an individual TAD (68), which may contribute to the stochasticity of gene regulation. Until the last decade, most models of gene regulation assumed that transcription factor binding was under near thermodynamic control (69). It is now clear that this assumption of adiabatic DNA binding is a suspicious one for bacteria and probably wrong for eukaryotes (70, 71). Clarifying whether nonadiabatic TAD folding events and other chromosomal transitions are big contributors to the stochasticity of gene transcription should loom large in the agenda of cellular biophysics (72, 73). TAD transitions, as well as changes in chromatin type causing phase separation may be super-nonadiabatic: they may be slower than most regulatory circuit responses. Types even might be preserved through the process of mitosis as true epigenetic features. To study this very long timescale question, analysis will be needed into how the effective energy landscape of chromosomes changes through the stages of development.

We hope this perspective shows that energy landscape theory both answers some old questions about chromosomes and provides a framework for asking new questions about how the genome’s story is acted out in living cells.

Author Contributions

B.Z. and P.G.W. wrote the article.

Editor: Tamar Schlick.