"Thermodynamic Pathways to Genome Spatial Organization in the Cell Nucleus".

Mario Nicodemi ^{1, 2, @} and Antonella Prisco ³

¹ Department of Physics and Complexity Science, University of Warwick, Coventry, United Kingdom
² INFN Naples, Naples, Italy
³ CNR Institute of Genetics and Biophysics Buzzati Traverso, Naples, Italy

^@ Corresponding author
mario.nicodemi@na.infn.it

Publication Information: Received: May 23, 2008; Accepted: December 9, 2008; Editor: Steven D. Schwartz.

The architecture of the eukaryotic genome is characterized by a high degree of spatial organization. Chromosomes occupy preferred territories correlated to their state of activity and, yet, displace their genes to interact with remote sites in complex patterns requiring the orchestration of a huge number of DNA loci and molecular regulators. Far from random, this organization serves crucial functional purposes, but its governing principles remain elusive. By computer simulations of a statistical mechanics model, we show how architectural patterns spontaneously arise from the physical interaction between soluble binding molecules and chromosomes via collective thermodynamics mechanisms. Chromosomes co-localize, loops and territories form, and find their relative positions as stable thermodynamic states. These are selected by thermodynamic switches, which are regulated by concentrations/affinity of soluble mediators and by number/location of their attachment sites along chromosomes. Our thermodynamic switch model of nuclear architecture, thus, explains on quantitative grounds how well-known cell strategies of upregulation of DNA binding proteins or modification of chromatin structure can dynamically shape the organization of the nucleus.

Within cell nuclei, genomes have a complex organization in space spanning different scales. Chromosomes tend to form a set of distinct territories and, at a smaller level, are folded in higher-order structures, while a variety of physical intra- and interchromosomal interactions between specific DNA sequences take place (1,2,3,4,5,6). While structures can be formed by tethering specific DNA segments to scaffolding elements, such as the nuclear envelope, DNA-DNA contacts and chromatin loops are an ubiquitous organizational feature extending up to hundreds of kilobases, and relocating, for instance, genes to substantial distances outside of their territory. Intriguingly, relative positions of territories, as well as of DNA sequences within a territory, have a probabilistic nature dynamically changing with cell type and cell cycle phase. Yet, stable, nonrandom patterns are established, fundamental to genome regulation, as disruptions relate to serious diseases, most notably, cancer (1,2,3,4,5,6). Remarkably common features are shared in chromatin organization processes, but the underlying principles of their control in space and time are still largely mysterious (3).

Although there is evidence that far-apart DNA sequences, even on different chromosomes, can come together by interacting with molecular factors, the mechanisms whereby they do so and higher-order structures and territories that arise are still largely mysterious. One of the scenarios proposed to explain the establishment of contacts between DNA elements is the so-called random collision picture (see, e.g., (7)) whereby chromatin flexibility allows factors bound to one sequence to randomly contact factors bound to surrounding chromatin. Although active mechanisms of directed motion have been described (see, e.g., (8)), diffusion-based mobility is indeed a prevailing mechanism that delivers molecular complexes to their specific nuclear targets (9). So, loops could be formed when a diffusing factor succeeds in bridging two chromosomal sites as a result of a random double encounter, whereby the molecule by chance encounters its first binding site and then, by chance, the second one. Yet how such loops persist beyond the initial random collision is totally unclear (7,10) and many questions remain open: how strong are the bonds required to hold in place whole chromosomal segments? How are stochastic encounters coordinated in space and time for a functional purpose by the cell? Can higher-order structures and territories spontaneously arise from them? Here, by use of a polymer physics model we propose a scenario to answer such questions.

Sequence-specific DNA-binding molecular factors have emerged as critical regulators of chromatin interactions in the nucleus (1,2,3,4,5,6) and some of them are encountered in a variety of cases, as for instance SATB1 (11), Ikaros (12), PcG (13), and CTCF Zn-finger proteins, the latter known to mediate also interchromosomal contacts (14,15,16,17). In some cases, a combination of factors is required to induce looping, as in the example of the erythroid transcription factor GATA-1 and its cofactors at the b-globin locus (18,19). Analogously, GATA-3 and STAT6 cooperation has been proposed to establish long-range chromatin interactions at the TH2 cytokine locus (20). Transcription factories themselves, i.e., local high concentrations of Pol II, have been proposed to act as hubs in the formation of loops and the colocalization of distant genes, even outside chromosome territories (5,21,22). In the last few years, protein-DNA interactions that occur in vivo have been probed by innovative genome-wide techniques leading to the description of thousands of binding sites for DNA binding proteins (23), and systematic approaches to measuring their binding energy landscapes are being developed (24). DNA binding proteins typically exhibit a number of target loci, which can be found clustered in groups. Their DNA chemical affinities are in general found in the weak biochemical energy range, E_X~1/ 15 kT (24,25,26,27,28) (k is the Boltzmann constant and T room temperature). Although in most cases only qualitative information is available, details on binding energies and DNA locations have been clearly described for a number of examples (24,25,26,27,28). Initial works on bacteria have shown that DNA binding proteins can have hundreds of DNA sites with affinities in the range 2/15 kT (26). In yeast, more recently, the landscape of binding energies and loci has been explored by advanced computational biophysics methods: the distribution of their binding energies spans a range of ~10 kT, and they can have hundreds of DNA binding sites across the genome as well (28). Similar ranges in binding energies have been found in higher eukaryotes, including mice and humans (23,24,25), where common examples exist of proteins with thousands of DNA target sequences.

DNA-DNA interactions mediated by molecular factors are being extensively mapped, revealing a complex network of intra- and interchromosomal interactions (29). Clusters of binding sites of SATB1 (11,30), and zinc finger class proteins CTCF (14,31,32), Ikaros (12), and GATA-1 (33) were found in a number of regions involved in DNA cross talk. An important example is the cluster of CTCF binding sites responsible for X chromosome pairing, at the onset of X inactivation, located at the Xist/Tsix locus where, in a few-kb-short sequence, a group of 100 binding sites, each 20-b long, is found (17,31). Expansion of the nuclear volume leads to the disassembly of several nuclear compartments (34), which might suggest that specific concentrations of macromolecules are required for the self-assembly of nuclear structures. Loss of specific interchromosomal DNA-DNA contacts has been described after a marked reduction, for instance, of the amount of CTCF (15,17). Changes in the concentration of heterochromatin proteins, e.g., HP1 (35,36), are also known to affect the organization of genomic DNA (37).

The conformational properties of chromosomes have been investigated by using polymer models in the past (38,39,40,41,42,43,44,45,46,47,48,49). The chromatin fiber was modeled as a random walk in a confining geometry (38), and the possibility was considered to include giant loops, of 3 Mb, departing from its backbone to describe folding at different scales (RWGL model (39,40)). The multi-loop-subcompartment MLS model (41,42) aimed to represent rosette structures, with 120 kb loops, like those experimentally observed. To describe the radial arrangement of chromosome territories in human cell nuclei, a model was proposed (43) where each chromosome is approximated by a linear chain of spherical 1 Mbp-sized chromatin domains. Adjacent domains are linked together by an entropic spring and by an effective excluded volume potential, while to maintain the compactness of chromosome territories a weak potential barrier around each chromosome chain was also included. Recently, the random loop polymer model (47) was introduced to explain, at the same time, the experimentally observed presence of loops of different scales and the leveling-off of the mean-square distance between two beads of the chain at genomic distances at >12 Mb. Several other chromatin features have been successfully explored by computer simulations, including nucleosome interactions (44), packing (45,46), and molecular assembly (48,49), providing a vivid description of the geometry and conformational properties of chromatin as observed in experiments.

Here, by investigation of a polymer physics model inspired by the above biological scenario, we discuss how architectural patterns spontaneously arise from the interaction of soluble-binding molecules and chromosomes. Our model shows that thermodynamics dictates pathways to complex pattern formation: loops, colocalization of distant sequences, chromosomal domains, structures, and territories spontaneously organize as stable thermodynamic states when specific threshold values in molecule concentrations or their affinity to DNA sites are exceeded. By regulation of expression levels and modification of DNA targets, the cell can, thus, act on thermodynamic switches (50,51) to reliably control its genome organization in space and time.

Theoretical Model:

To describe a system made of a chromosome and its binding molecules, we consider an established model of polymer physics (52,53): the chromosome polymer is modeled as a Brownian self-avoiding walk (SAW) of n non-overlapping beads, and soluble molecules as Brownian particles having a concentration, c (Fig.1). A fraction, f, of polymer sites can bind the diffusing molecules, with a chemical affinity EX in the weak biochemical range (see Methods for details). Here, for sake of simplicity, binding sites are uniformly interspersed with nonbinding regions along the chain. Each molecular factor can simultaneously bind many a site on the polymer, a feature that reflects the presence of multiple DNA binding domains in a number of regulatory proteins (e.g., CTCF). Mediating molecules with only one DNA binding site, that are able to interact with each other, could be also considered; since a group of linked molecules can be represented, in the model, as just one mediator, the picture is unchanged. The equilibrium thermodynamic properties of such a system were determined by extensive Monte Carlo simulations (53,54).

Figure 1: Model.

Figure 1: Model.

The figure shows two representative snapshots from our three-dimensional computer simulations. In the left panel, a self-avoiding walk (SAW) polymer is shown, as it floats randomly within the assigned volume without forming stable loops. In the right panel the volume also contains a concentration c = 0.04% of Brownian molecules (yellow) having an affinity EX = 4 kT for a fraction f=1/3 of the polymers beads (shown in a darker shade). As molecules can bind more than one polymer site, loops can be formed. However, they are stable, and confine the polymer in a closed territory (as in the case shown here), only if c is above a threshold value (see Fig.2). The SAW chains shown here comprise n = 64 beads.

In our Monte Carlo computer simulations (53) molecules and polymers diffuse in a cubic lattice having a linear size L, and its spacing, d0, sets the space unit. For computational purposes, we mostly consider lattices of linear size L = 32, although we tested our results up to L = 128. SAW polymer beads have a diameter, d0, and each bead in a chain is on a next or nearest next-neighboring site of its predecessor. Molecules (of size d0) are also subject to Brownian motion. When neighboring a binding site of a polymer, molecules interact with it via an effective energy, EX. According to the studied case (see Results ), up to six distinct sites (i.e., the nearest neighbors in a cubic lattice) on the same chromosome, or alternatively two sites on different chromosomes, can be bound at the same time.

Our schematic model is a coarse-grained description of a real polymer and, since by now we mostly focus on the description of a general conceptual framework, beads only represent generic binding sites (they could be a binding locus, the bases of specific binding sequences, etc.). In cases where detailed data on binding sequences and regulator chemistry is available, such information could be easily taken into account in the model to produce specific quantitative predictions. The role of interactions with, e.g., the nuclear membrane could be also included, but to make the message simpler, we decided not to discuss such an aspect here.

To obtain thermodynamic equilibrium configurations, the Metropolis Monte Carlo method was applied. Chromosome polymers are initially equilibrated in a random self-avoiding configuration obtained, in absence of binding molecules, by random displacements of single beads under the constraint that each bead in the chain is on a next or nearest next-neighboring site of its predecessor. Then molecules are inserted at random empty positions in the lattice to attain a given concentration. In the ensuing Metropolis Monte Carlo procedure, a sequence of states is generated by a Markov process (53), whereby a new position for a particle/bead is stochastically selected according to a specific transition matrix satisfying the principle of detailed balance, which, in turn, guarantees the convergence in probability of the sampled states to Boltzmann thermodynamic equilibrium distribution. The transition probability for a particle/bead to diffuse to a neighboring empty site is proportional to the Arrhenius factor r₀ exp(-DE/kT), where DE is the energy barrier in the move, k the Boltzmann constant, and T the temperature (53). The lattice has periodic boundary conditions to reduce boundary effects.

In a Monte Carlo lattice sweep, every particle and bead in the system, randomly selected, is updated on average once. Our simulations run for up to 10⁹ Monte Carlo lattice sweeps as the number of decorrelation steps from an initial configuration can be as large as 10⁵. The achievement of stationarity was monitored by checking the dynamics of different quantities, such as the system gyration radius, the distance between two polymers, the system energy, and the number of particles attached to polymers. Once equilibrium is reached for all these quantities, thermodynamic averages are calculated by considering only configurations having a distance larger than the decorrelation length. Finally, averages are also performed over up to 2048 runs from different initial configurations. Confidence intervals are calculated as squared deviations around these averages (53); they are indicated in our figures by the size of the used symbols.

Our code has two core routines, well described in Binder and Heermann (53): the lattice gas spin-exchange Metropolis routine for particle displacement, and the SAW routine. Several means were considered to avoid algorithmic errors, as those suggested by Binder and Heermann (53). Each different routine in the code was tested independently. For example, the routine generating the evolution of the SAW chain was tested by checking the behavior of the calculated average gyration radius, R_g, against the chain length, n, and the power law R_g~nⁿ with an exponent n ~ 0.6, well established in the literature (52,53), was recovered. Another internal test was to show that other geometric quantities, such as the chain end-to-end distance, did scale in the same way as R_g.

Real chromosomes differ in size (i.e., n) and arrangement of their binding sites. Such differences affect their specific behaviors, but the general picture we aim to depict here is not altered by changes to the selected values of these parameters (e.g., n and L). To make computation time feasible, we mostly use n = 64, but we tried n as large as 128. The robustness of our model is well established in polymer physics (52,53), and to check the effects of finite size scaling, we explored changes of the polymer chain length in the range
n e {16,32,64,128} (see Results ).

Results:

Intrachromosomal interactions, loop, and territory formation

We first discuss how a chromosome can fold up in loops within a territory with a specific spatial conformation by interacting with soluble molecules, and how the process can be controlled by the cell (see Fig.1). The folding state of the polymer is illustrated by its squared radius of gyration, R_g² (52): , where ri is the position of bead i  {1, , n}, and N a normalization constant (here N equals the average squared gyration radius of a randomly floating SAW chain of size n). The value Rg represents the radius of a minimal sphere enclosing the polymer: it attains a maximum when the polymer is loose and randomly folded, and a minimum when loops enclose it in a compact lump.

In presence of a given concentration of molecules, loops could be created by chance when a particle bridges a couple of chromosomal sites having a nonzero affinity, EX. Fig.2 (left panel) shows, indeed, that R_g² attains a small plateau value when EX is large enough (say above the inflection point, E_tr, of the curve R_g²(EX)): bridges are thermodynamically favored and the polymer takes a compact looped territorial conformation, as seen in a typical snapshot from computer simulations depicted in Fig.1 (right panel). The system behavior, however, switches for EX < E_tr, since R_g² keeps its maximal value corresponding to a fully open polymer floating in space (see Fig.1, left panel) and no stable loops are formed. The folding level also depends on factors such as concentration of molecules, number, and location of DNA binding sites (see below).

Figure 2: Thermodynamic switches for intrachromosomal contacts and loop formation.

Figure 2: Thermodynamic switches for intrachromosomal contacts and loop formation.

The equilibrium average gyration radius, R_g², of the model polymer pictured in Fig.1, depends on the affinity, EX, of its binding sites for a set of molecular factors, on the concentration, c, of those factors, and on the fraction, f, of polymer beads which can bind molecules. R_g represents the radius of a sphere enclosing the polymer: it has a maximum (R_g² = 1 in our normalization) when folding is random and a minimum when the polymer loops on itself in a lump (the horizontal red line is the radius of a compact sphere formed by the polymer). In the left panel, R_g² is shown as a function of EX, for a given value of c and f (here c = 0.04%, f = 1/3). For EX below a threshold value, , R_g² is 1 and the polymer is on average open. For EX > Etr, R_g² collapses, as the polymer forms a looped territory. In the central panel, R_g² is shown as a function of c, for a given EX and f (here EX = 4 kT, f = 1/3). In addition, in this case a threshold effect is observed ( c_tr~0.01%), although a broader crossover region exists where the level of folding can be tuned. The right panel shows the sharp threshold of R_g² as a function of f ( f_tr~0.1, here c = 0.04%, EX = 4 kT), illustrating that only in presence of multiple sites (i.e., above f_tr) the polymer can be folded in loops. In all the above cases, loops are thermodynamically stable only above the threshold values, as a consequence a phase transition occurring in the system. By tuning affinities/concentrations, the cell can act, thus, on a thermodynamic switch to form and release loops and territories.

The above results have an intuitive basis: if EX is small, the half-life of a randomly formed bridge is small and polymer segments on average float away; the higher EX, the higher the number of bound molecules and, thus, of bridges which reinforce each other and stabilize the conformation, as multiple bonds should be simultaneously broken to release a loop. Our physics model reveals, in particular, that a precise threshold marks the switch between the two regimes; E_tr corresponds to a thermodynamic phase transition (55), as discussed later on. This picture illustrates on quantitative grounds how chromatin modifications, such as DNA methylation or post-translational modifications of DNA binding proteins (well described cell strategies to change genomic architecture), can result in dramatic, switchlike, effects.

In a different thermodynamic pathway to loop formation, the cell can regulate the concentration, c, of binding molecules. The plot of Rg2(c) (Fig.2, central panel) shows how c affects the compaction state of the polymer. When c is below the threshold, c_tr, R_g²(c) has a value corresponding to random folding, while >ctr, it decreases toward its looped state value. A broad crossover region is found at ~c_tr, revealing that R_g², which can be envisioned in our example as the radius of the territory, can be tuned across a range of values. Therefore, the regulation of a DNA binding protein concentration (a typical event in cellular behavior) can act as another switch to reliable assembling of genomic architectures.

Finally, we find (Fig.2, right panel) that a minimal threshold in the number of polymer binding sites (or in their fraction f) is required for stable looping/territory formation. Conceptually, the case of a polymer with a low number of binding sites is equivalent to the case of a polymer with many binding sites in the presence of a limiting concentration of mediators (Fig.3). The function R_g²(f) indicates that a thermodynamic switch to DNA compaction resides in the potential to obliterate/restore a fraction of sites via chromatin modifications that abolish binding of the relevant regulatory molecule. Intriguingly, the presence of a thermodynamic threshold in f could relate to the experimental observation that multiple binding sites for mediators have been found at chromosome interaction loci and looping points (e.g., CTCF-mediated interactions). Importantly, in our model we find that the threshold value, f_th, is a strongly decreasing function of the binding energy, EX. This can be expected as, for an above threshold mediator concentration, c, the overall binding energy linking two polymer strands is ~f EX; so an increase of EX would correspond to an inversely proportional reduction of f_tr.

Figure 3: Scaling properties.

Figure 3: Scaling properties.

(Lower panel) The average gyration radius, R_g, relative to polymer model considered into the left panel of Fig.2, is plotted as a function of the polymer chain length n. The picture shows the ratio R_g²(n)/R_g²(64) (since n = 64 is the reference case dealt with in the rest of the article) for n = 16, 32, 64, 128. In the phase where the polymer is open, i.e., for EX= 1kT < E_tr (see left panel of Fig.2), the average gyration radius, Rg(solid circles), scales with n as a power law Rg~nⁿ with an exponent  n~0.6 (52,53) (superimposed fit, dashed line). In the looped phase, i.e., for EX = 4 kT > E_tr, R_g (open circles) scales as n^1/3 (superimposed fit, long dashed line), showing that the polymer is lumped in a compact conformation. (Upper panel) The threshold energy, E_tr, relative to the left panel of Fig.2, is a function of the polymer chain length n. Here we plot the ratio E_tr(n)/E_tr(64) (where E_tr(64)~3kT ). The superimposed fit is E_tr(n) = E_tr + A/n^B , where E_tr(n) is the threshold energy for a chain of size n, E_t ~0.96E (64),  the extrapolated value for an infinitely long system, , (A~0.47E_tr(64)and B~0.5 a fitting coefficient and exponent.

The above-described thermodynamic switches define a robust regulatory mechanism as seen in the phase diagram of Fig.4, reporting the equilibrium state of the chromosome (open versus looped) in a wide range of EX and c values (for a given f). In particular, Fig.4 shows that the threshold value E_tr(c) (dashed line) required for loop assembly decreases as c increases and can be as weak as a hydrogen bond. In the cell, the possibility to drive looping by use of sites with even low binding energy for their soluble ligands could be important to prevent polymers from getting stuck in topologically unacceptable entanglements or ectopic associations, since each single low energy bond can be easily broken for adjustments.

Figure 4: Phase diagram.

Figure 4: Phase diagram.

The state of the polymer/chromosome (see Fig.1) at thermodynamic equilibrium is summarized by this phase diagram in a range of values of weak biochemical affinities, EX, and concentration, c, of its binding molecules (here f = 1/3). When EX and c are below the transition line, E_tr(c) (open circles), the polymer is open (as sketched in the inset) and no stable loops can be formed. Above threshold, instead, the system enters the region where the polymer is folded and looped on itself.

The threshold values in the (c, EX, f) space (see Fig.4), related in polymer physics to the chain -point (52), correspond to a phase transition occurring in the system when one of two competing thermodynamics mechanisms prevails: entropy, S,  which favors loose random folding; or energy, E, which increases when bonds between molecules and DNA sites are established by loop formation. The system spontaneously tends (as it is finite-sized (55)) to select the state where its free energy, F(c, EX, f) = E - TS, is minimized. More precisely, the chromosome conformation has a specific stochastic distribution (having a width which can be very narrow) following from Boltzmann thermodynamics weights (55).

Scaling behavior of the model

As molecule binding regions on cross-talk loci of real chromosomes have variable sizes, n, we explored the scaling behavior of our system by varying the polymer chain length in the range ne  {16,32,64,128}, for the above value of the containing box size L. The reference case considered previously, and in the rest of the article, has n = 64, which is comparable to values included in similar studies (41,42,43).

We investigate, in particular, how the average gyration radius, R_g, and the threshold energy, E_tr, depend on n. For a matter of clarity, we refer to the case discussed in the left panel of Fig.2, but similar features are found for the other cases presented in our article. We, thus, consider a system with c = 0.04% and f = 1/3, and discuss first the case where EX = 1 kT, i.e., the phase where the polymer is open (see Fig.2, left panel). Under these circumstances, as shown in the lower panel of Fig.3, R_g scales with n as a power law, R_g ~ nⁿ, with an exponent n ~ 0.6, which is in agreement with the random SAW scaling laws (52,53). Conversely, for EX = 4 kT, i.e., in the looped phase, R_g scales as n^1/3 (see lower panel of Fig.3), showing that the polymer is lumped in a compact conformation (1/3 is the inverse of the Euclidean dimension of the system). The threshold energy E_tr has also a comparatively simple behavior with n and appears to saturate to a finite value for large n. For instance, the threshold energy defined in the left panel of Fig.2 (where E_tr (64)~3kT  ) can be well fitted by a power law in n (see upper panel of Fig.3): E_tr(n) = E_tr + A/n^B, where Etr(n) is the value for a chain of size n,  E_tr the fitting value for an infinitely long chain, A and B a fitting coefficient and exponent (we find E_tr ~0.96E_tr(64), A~0.47 E_tr(64) ,  and B~ 0.5).

Similar properties are found for the other quantities discussed in this article. These checks outline the robustness of the picture discussed above and also support the idea that it is not an artifact of discretization, as a system in the continuum limit, i.e., on a finely divided lattice, should have an analogous behavior.

Interchromosomal segment interactions

The mechanisms that drive other layers of spatial organization, including the colocalization of DNA sequences belonging to different chromosomes (56) and the relative positioning of chromosomal territories (1,2,3,4,5,6), can be shown to be very similar to those inducing stable loop formation within a single chromosome. Concentration/affinity acts in these cases as a thermodynamic switch for segment colocalization and for chromosome positioning in a map.

To such an aim, in an extension of the model described above, we now investigate the thermodynamic state of two SAW chains (representing either two distal sequences on the same chromosome or sequences on distinct chromosomes) with a fraction f of binding sites (periodically placed) for a concentration, c, of molecules having an affinity, EX, to both of them (Fig.5); for simplicity, each molecule can bind once either polymer. The relative polymer positioning is given by their squared distance:  d² =  1/(2n²D)(r_i⁽¹⁾ - r_j⁽²⁾  )²
, where  r_i⁽¹⁾  (respectively, r_i⁽²⁾) is the position of bead i in chromosome 1 (respectively, 2), and D a normalization constant (here  D is equal to the average square distance of two independent random SAW chains). The average value of d² is maximal when polymers float independently (i.e., d² = 1 in our normalization) and decreases drastically when all or parts of the chains become colocalized.

Figure 5: Two snapshots are shown from computer simulations of our two polymer model.

Figure 5: Two snapshots are shown from computer simulations of our two polymer model.

In the left panel, the polymers float independently within the assigned volume. In the right panel, the volume also includes a concentration, c = 0.3%, of molecules (yellow particles), which can bind simultaneously each polymer once at anyof their specific loci (darker sites, here in a fraction f= 1/2 with affinity EX = 4 kT). When c is above a threshold value (see Fig.6), as in the case shown, thermodynamically stable bridges can be formed between the polymers, which spontaneously tend to pair parts of, or all of, their chains.

Regulation of EX can induce formation or release of stable physical contact between the polymers. Fig.6 shows that when EX is below a threshold, E_tr, their equilibrium distance, d², has the same value found for two noninteracting Brownian SAW chains (i.e., d² = 1). This is the random phase where chromosomes move independently. By thermodynamics mechanisms an effective attraction between the polymers is, instead, established when EX > E_tr: physical contact is stable and d² drastically decreased, as the system enters the colocalization phase. The equilibrium distance is a function of c as well (see Fig.6): when c is below a threshold value, c_tr, a random distance is found between chromosomes (i.e., d² = 1). Colocalization is spontaneously attained, instead, when c increases, as d²(c) approaches a plateau with a much smaller value. Finally, for a given c and EX, colocalization can be achieved only if the number of binding sites along the polymers is above a sharp threshold value, as shown in Fig.6 where d²(f) is plotted.

Figure 6: Thermodynamic switches for interchromosomal contacts.

Figure 6: Thermodynamic switches for interchromosomal contacts.

The equilibrium average distance, d², of the two polymer model pictured in Fig.5, is a function of the affinity, EX, of their binding sites for diffusing molecules; of the concentration, c, of molecules; and of the fraction, f, of polymer binding sites. In the left panel, d² is plotted as a function of EX (here c = 0.3%, f = 1/2). When EX is smaller than a threshold, , d² is maximal (d²= 1 in our normalization) and the polymers float independently one from the other. For EX > Etr, d² drastically decreases, as the polymers are spontaneously colocalized. In the central panel, d² is shown as a function of c (here EX = 4 kT, f = 1/2) and a threshold appears as well (c_tr~0.07%), surrounded by a crossover region. In the right panel, the sharp threshold of d² as a function of f is shown (, here EX = 4 kT, c = 0.3%): only multiple binding sites, above f_tr, can achieve polymer colocalization. The mechanism driving polymer colocalization is an effective reciprocal attraction of thermodynamic origin, related to a phase transition: below threshold, molecules bridging by chance the polymers do not succeed in holding them in place; above threshold, bridges are thermodynamically stabilized. Molecular mediators act, then, as a thermodynamic switch to spontaneous formation and release of polymer stable contacts.

As with loop architecture within a chromosome territory, the average distance of chromosome pairs can be controlled via thermodynamics mechanisms. The spatial association is attained when a phase transition line is crossed, corresponding to the point where entropy loss due to chain pairing is compensated by energy gain as both polymers are bound: the lower the EX, the higher the concentration, c, required.

Assembly of chromosome territorial maps

Within the above picture, the relative positioning of chromosomal loci and territories can be understood by similar arguments. As an example, we considered (see inset in Fig.7) the case with three SAW chains (n = 64) having each a fraction, f, of binding sites (f = 1/2, EX = 4 kT): the sites on polymer 1and 2 interact with a molecular factor (concentration c₁₂) which can bind once either chain; polymer 2 and 3 bind a different molecular factor of concentration c₂₃ (for definiteness, we only discuss the case where c₁₂ = c₂₃ = c). To illustrate the important effects of physical interference between chromosomes, in this model all molecular factors compete for the same sites on polymer 2. For the built-in symmetry, polymer couples 1-2 and 2-3 behave similarly and have, on average, equal relative distances  as a function of c (see Fig.7). Yet, since polymer 1 and 3 physically interfere when bridging with 2, in a competition for its binding sites, their distance is larger than the one found in the case with only two polymers under similar conditions (i.e., same c, EX, f and system size). The distance between 1 and 3, d₁₃, is in turn larger than d₁₂ = d₂₃ because there is not a direct interaction. The three chromosomes, thus, spontaneously find their position to form a (isosceles) triangle having sides of predefined length (d₁₂, d₂₃, d₁₃).

Figure 7: Territorial maps.

Figure 7: Territorial maps.

The relative positions of three polymers can be regulated by the concentration of specific molecular factors. (Inset) A configuration is shown from our computer simulations of a three polymer model. A specific molecular factor can bind polymers 1 (pink) and 2 (blue), while a different factor binds polymers 2 and 3 (orange). Both molecular factors have here a concentration c = 0.13% (EX = 4 kT, f = 1/2), but they are not shown for clarity. (Main panel) The average distance between polymers 1 and 2,  (squares), decreases as a function of c (the distance between 2 and 3 equals , and is not shown). As an indirect effect of the attraction within pairs 1-2 and 2-3, the distance between 1 and 3,  (diamond), decreases as well, remaining, however, above . The three polymers, thus, tend to form a triangle with two short equal edges (corresponding to d₁₂ and d₂₃) and a longer edge (i.e., d₁₃). In general, by tuning c, EX, and f, a variety of configurational patterns can be spontaneously attained. Notably, since polymers 1 and 3 compete for bridging the sites of polymer 2, they physically interfere and  is larger than in the case of an isolated couple (yellow lower line, from Fig.6). A proper spatial organization of chromosomes in territories and within territories could also help in minimizing physical interference and entanglement.

Different patterns of relative positions can be attained by tuning the concentration/affinity switches, as the system architecture self-organize via thermodynamics pathways, funneling the interaction between sets of DNA binding sites and matching molecular mediators. When the number and length of chromosomal segments increase, the dynamics of the system to equilibrium can be slowed down by physical hindrance. Thus rises the speculation that the spatial organization of chromosomes in distinct territories and within territories (along with other mechanisms, e.g., the action of topoisomerases) may also serve the purpose of a faster and better control of their interaction and function, by reducing undesired entanglements
(? cancer, Editor).

Discussion:

Within the cell nucleus, in a striking example of self-organization, an astonishing number and diversity of DNA loci and molecular mediators are spatially orchestrated to form a complex and functional architecture involving regulatory cross talk between distant sites. We propose a simple conceptual framework, a thermodynamic switch model of nuclear architecture, to understand some of its general features, namely (4):
1), how a chromosome can fold up into a territory and how its looping is dynamically controlled by binding molecules;
2), how chromosomes interact and establish their relative positioning;
3), what the regulatory principles are; and
4), the origin of the stochastic character of territorial maps.

Our model consists of a system of SAW polymers interacting with soluble molecular mediators. By use of statistical mechanics, we have shown that thermodynamics dictates pathways to complex pattern formation, via mechanisms such as thermodynamic switches (Fig.8). This supports, on quantitative grounds, the idea that a variety of intra- and interchromosome interactions can be traced back to similar mechanisms. Looping and compaction, remote sequence interactions, and territorial-segregated configurations correspond to thermodynamic states selected by appropriate values of concentrations/affinity of soluble mediators and by number and location of their attachment sites along chromosomes. After proper concentrations/affinities are set, the organization proceeds spontaneously with no energetic costs as the resources required, e.g., to rearrange even whole chromosomes, are provided by the surrounding thermal bath. Our picture explains, thus, how well-described cell strategies of upregulation of DNA binding proteins or modification of chromatin structure can shape the genomic architecture and produce DNA colocalization and territories according to thermodynamically driven nonrandom patterns.

Figure 8: Schematic illustration of thermodynamic switches and their effects at different levels of system organization.

Figure 8: Schematic illustration of thermodynamic switches and their effects at different levels of system organization.

(Top panel) The assembly of intrachromosomal contacts and loops is thermodynamically possible only when the concentration/affinity of binders (circles) exceeds precise threshold values. At that point, previously randomly and independently diffusing molecules and chromosomes spontaneously generate an organized pattern, in a process reversible by downregulation of the switch. Specific conformations can be attained by site specificity of a set of molecular mediators.

(Bottom panel) Similar threshold and self-organization mechanisms act for establishing contact between remote loci and, at a higher scale, relative positions of territories. A variety of patterns, encoded in the location of a number of binding sites along chromosomes, can be precisely selected via thermodynamics effects by a combinatorial use of a set of molecular mediators (rectangles).

Testable quantitative predictions are shown on the biological effects of alterations of genomic DNA sequences (such as deletions, insertions, chemical changes, etc.) and of their molecular mediators (concentrations, binding energies, etc.). In particular, the model highlights the fact that, at above-threshold values of concentration, the interaction with low affinity molecular factors may be sufficient to drive the compaction of chromosomes into territories, and shows that the interaction of chromosomes with soluble mediators has the potential to impart a probabilistic relative arrangement to chromosomes. Our analysis reveals that molecular factors that act as bridges between two chromosomes may not only have the effect of pulling those close to each other, but may also displace noninteracting chromosomes, so that these are farther away from each other than the random distance. This result is thought-provoking in the light of experimental data (56) showing that disruption of transcription can lead either to an increase or to a decrease of chromosome intermingling among specific couples of chromosomes, depending on which two chromosomes you look at. Allele-specific, parent-of-origin specific, and expression-specific DNA-DNA interactions have also been described (15,16,57,58,59). In this context, our analysis could explain how imprinting and other allele-specific protein-DNA interactions may have the capacity to address homologous chromosomes to two different regions of the territory map.

A rough estimate of threshold molecular concentrations in real nuclei can be made from our predicted concentration values: here c is the number of molecules per lattice site, so the number of molecules per unit volume is c/d₀³, where d₀ is the linear lattice spacing constant. The molar concentration  is obtained by dividing by the Avogadro number N_A. Note that threshold concentrations depend on the binding energy EX (see, e.g., Fig.4). For sake of definiteness,however, we can consider the case with EX ~ 2 kT (see Fig.4), where threshold concentrations are c = 0.1%  0.01%. Under the rough assumption that d₀ is a couple of orders-of-magnitude smaller than the nucleus diameter (i.e., d₀ ~ 10 nm), a threshold molar concentration would be r~ 0.1 /  1 mmole/liter, which is consistent with typical experimental values of nuclear protein concentrations (60,61). Such estimation is very rough, but may help to further bridge this study with biological investigations.

Starting from experimental results showing that chromatin fiber at large genomic distances, >12 Mb, exhibits a leveling-off of the mean square distance between two DNA sites, a Gaussian random loop polymer model was recently proposed (47). To explain these observations, the model proposed that a given number P of couples of distant beads, randomly selected along the chain, are bound by an harmonic potential of amplitude k. The model investigated the mean distance between sites and the size of loops, and showed that the presence of random loops on all length scales explains the leveling-off of the mean-square distance. That model is close in spirit to this work where cross interactions of a fraction, f, of DNA sites are mediated by the binding of molecular factors and by the formation of bridges of energy EX. In our case, the number of interacting site couples also depends on the concentration of mediators, c, and their interaction is short-ranged. Nonetheless, the number of interacting site in our model would correspond, in the notation of Bohn et al. (47), to a P, which is a (nontrivial) function of c. These considerations can illustrate the agreement between the discovery in Bohn et al. (47) of the leveling-off of the mean-square distance and our finding, for instance, that for c above threshold, the polymer gyration radius does not attain the (self-avoiding) random walk value but saturates to much smaller values.

In real cells, passive and active regulatory mechanisms can cooperate, adding further layers of complexity (1,2,3,4,5,6), while the list of molecules mediating chromatin organization is likely to include dedicated structural proteins, RNAs and, e.g., the transcription, replication, or repair machinery (21,62,63). In our picture, specificity of interactions is obtained by specific molecular mediators binding to specific loci, while other general molecules could help the process. In the arrangement of specific binding sites along chromosomes and scaffolding elements, a variety of spatial patterns can be encoded (43) on an evolutionary timescale. Within a cell, patterns could be then dynamically selected by the combinatorial use of a set of mediators via the ineluctable, yet probabilistic, laws of thermodynamics (45).

Acknowledgments:

The authors declare no competing interests.
Work supported by MIUR-FIRB grant No. RBNE01S29H, Network No. MRTN-CT-2003-504712.

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In this beautiful and original formulation of the biophysics of the chromatin microfibril, Mario Nicodemi and Antonella Prisco have demonstrated the importance of the level or scale of analysis for each physical study. When analyzing the atom, the photon, the electron, or the sub-nuclear particles, we need to focus on our target level or scale  within its system. Within a given animal, the organ, the cell, the gene cluster, the chromosome, or the chromatin microfibril delineate distinct fields of investigation within the cell nucleus. 10 nm chromatin microfibrils are found within two types of chromatin complexes, active euchromatin and silent heterochromatin. This functional separation between two chromatin physical states has profound consequences for gene transcription and gene replication. The silent heterochromatin of the spermatozoon  undergoes a profound transition during oocyte fertilization, resulting in the activition of many embryonic genes into the euchromatin state.

This physical transition appears to result in a marked increase in the entropy of the chromatin microfibrils, their genes, and their transcripts. It may play a role in the apparent non-reversibility of the neoplastic process, unless met by the neg-entropic effects of microRNA molecules during the reversion process. Then, such a biophysical reversion to embryonic gene silence may support a successful RNA clinical therapy within embryoma patients.

It may, in fact, be the beginning of an effective therapy for all cancers, driven by  the study of  the biophysics of the cancer cell, following the second law of thermodynamics, DG = DH -  TDS, where:
DG is the reaction change in free energy, DH is the change in reaction enthalpy, T is the reaction absolute temperature, and DS is the change in reaction entropy.

Maximum entropy defines the reaction steady-state equilibrium,  DS = (DH - DG) / (T).
DS  is maximized within all isolated  chemical  reactions. The living cell, of course, does not permit isolated chemical reactions under normal conditions, but cancer is not a normal condition. Perhaps entropy should be minimized within cancer cells.

1. Hovsepian JA, and Frenster JH,  (2003b)
"Euchromatin as an Extensile Force within Mammalian Cell Nuclei".

2. Hovsepian JA, and Frenster JH,  (2003c)
"Bioassays of Isolated Nuclear RNA Species as Activators of DNA Transcription".

3. Frenster JH, and Hovsepian JA,  (2004a)
"Activator RNA Exchange during Interphase Chromatin Reprogramming".

4. Frenster JH, and Hovsepian JA,  (2004b)
"Ultrastructure  of Closed Loops within Euchromatin of Isolated Lymphocyte Nuclei".

5. Hovsepian JA, and Frenster JH,  (2004c)
"Reprogramming as an Approach to Neoplasms".

6. Hovsepian JA, and Frenster JH,  (2005a)
"Sense and Antisense during RNA Initiation of the DNA Transcription Bubble".

7. Frenster JH, and Hovsepian JA,  (2005b)
"Ultrastructure of Euchromatin Contact Points between the Closed Loops of Adjacent Interphase Chromosomes".

8. Frenster JH, and Hovsepian JA,  (2006a)
"Kissing Chromosomes and Paired Sense-Antisense RNA Synthesis".

9. Frenster JH, and Hovsepian JA,  (2006b)
"Activator RNA Initiation of the DNA Transcription Bubble".

10. Hovsepian JA, and Frenster JH,  (2006c)
"Chromosome-Chromosome Contact Points and Paired Sense-Antisense RNA Synthesis".

11. Frenster JH, and Hovsepian JA,  (2006d)
"DNase-I Ultrastructural Probe Sites and Kissing Chromosomes".

12. Rossi JJ,  (2007)
"Transcriptional activation by small RNA duplexes".

13. Frenster JH, and Hovsepian JA,  (2007a)
"DNA-DNA Tetraplex Model of Paired Sense-Antisense RNA Synthesis".

14. Frenster JH, and Hovsepian JA,  (2007b)
"Embryonic Gene Re-expression May Initiate Adult Neoplasms".

15. Okita K, Ichisaka T,  and  Yamanaka S, (2007c)
“Generation of germline-competent induced pluripotent stem cells”.

16. Frenster JH, and Hovsepian JA,  (Dec. 2008a)
"Models of successive levels of resolution during individual gene transcription".

17. Frenster JH, and Hovsepian JA,  (Dec. 2008b)
 "Micro RNAs and adult neoplasms of embryonic type".

18. Hovsepian JA, and  Frenster JH,  (2009).
"Genomic Models of Functional Embryomas within Adult Neoplastic Cells".

19. Frenster JH, and Hovsepian JA,  (2009).
"Functional Embryomas as a Result of Embryonic Gene Re-expression".

20. Frenster JH, and Hovsepian JA, ( 2010 )
"Cellular Dynamics of Embryomas within Adult Neoplasms".

21. Frenster JH, and Hovsepian JA,  ( 2010 )
"Analysis of Intra-Nuclear Entropy Changes during EMT Activation".

22. Hovsepian JA, and Frenster JH, ( 2010 )
"Heterochromatin - Euchromatin Transition within Functional Embryomas".

23. Frenster JH, and Hovsepian JA,  ( 2010 )
 "The Biophysics of the Cancer Cell".

1. Pathways within cell genomes involve a flow of information.

2. Information can flow by direct contact or by third parties.

3. Direct contact within whole genomes is difficult to regulate.

4. DNA-DNA direct contects are influenced by agents.

5. Nuclear agents include hydrophilic ionic and hydrophobic conforming ligands.

6. Third parties within genomes involve RNAs and proteins.

7.  RNAs and proteins are easy to regulate or reverse.

8.  Information can be shared, lost, or transformed.

9. System information can be hidden during system isolation.

10.  Local information can be permanently lost during system entropy.

http://www.embryomas.net/

Links to Current Research in Euchromatin:
Links to Euchromatin Activator RNA Reviews:
Links to Euchromatin Activator RNA Research:
Links to Ultrastructural Probes of DNase I-Sensitive Sites:
Links to RNA as a Therapeutic Agent:
Links to Hodgkin Lymphoma Immuno-Pathology:
Links to Activated T-Lymphocyte Immunotherapy:
Links to Medical Systems Biology:
Links to Selective Gene Transcription:
Links to RNA-Induced Epigenetics:
Links to RNA-Induced Embryogenesis:
Links to RNA and Biological Causality:
Links to Reprogramming and Neoplasia:

A Brief History of Activator RNA:

"Ultrastructural Probes of Active DNA Sites, and the RNA Activators of DNA".
(PowerPoint Presentation).

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