Fractal Dimension of Chromatin and Cancer Prognosis

KEYWORDS::

• cancer
• chromatin
• fractal
• fractal dimension
• nucleus
• prognostic factor
• self-similiarity

Fractals are self-similar structures, which exhibit similar features when examined at increasing magnifications [1]. Therefore, there is no characteristic scale for their description. They obey a nonlinear, power law relationship. Fractality can be quantified by the fractal dimension, which indicates how completely a fractal structure fills a space when we are using finer scales. The fractal dimension can be estimated from the slope of a linear regression in a log–log diagram, where the scale size is plotted against the values of a variable measured at different scales. In naturally occurring fractals, self similarity is sometimes flawed across the different scales. Therefore, the empirically determined points in the log–log plot do not exactly coincide with the ideal mathematical regression line. The goodness of fit between this ideal line and the measured points is another interesting feature that shows us how close the structure of the real object is to an ideal (mathematical) fractal [2,3].

The fractal concept, which was introduced by Mandelbrot [1], is ubiquitous in nature. The extension of its geometry towards the life sciences has improved our understanding of functional properties and dynamic physiological phenomena [4,5].

Fractal structures may be observed, for instance, in the anatomy of the vascular and pulmonary systems. Fractals are also very useful to properly characterize the complexity of macroscopic and microscopic anatomic structures, describing relevant design principles that underlie living organisms.

Fractality can be detected not only in objects but also in sequences of signals. Power laws are present in the regulation of blood pressure, ion channel kinetics, heart rate variability, allometric scaling growth, allosteric enzyme kinetics, metabolic rate in mammals, population genetics and the modeling of drug clearance [4,5]. A fractal organization has evolutionary advantages in many aspects. Their structures can be built by rather simple, iterative programs. Fractal branching is an efficient way for the construction of complex connections out of a simple recursive rule, which is applied across all scales, resulting in short distances for transport. Fractal foldings of membranes allow for the creation of a large surface area within a very small volume. Power law organization of physiological systems enables organisms to operate in a similar way at different scales. This increases the capacity of adaptation in case of changes in the environment [4].

Regarding the striking physiologic advantages and the ubiquity of self-similar organization in living organisms, it is not surprising that fractality can also be found in the organization of the genome and the epigenome. Several investigators demonstrated the presence of self similarity in DNA sequences [6–8]. In 1989, Takahashi created a fractal model of chromosomes and chromosomal DNA replication and stressed the importance of the fractal dimension as a measure of chromatin condensation [9]. Spinelli calculated the fractal dimensions of representative heterochromatic sequences and proposed a theory of heterochromatin as a system evolving in a self-organized manner at the edge of cellular and environmental chaos [10].

Recent experimental studies demonstrated striking evidence for the fractal organization of chromatin. Lebedev et al.[11] investigated the fractal nature of chromatin organization in intact interphase chicken erythrocytes by small-angle neutron scattering. The spectra revealed a constant fractal dimension of the protein component, while the DNA organization was biphasic, with the fractal dimension slightly higher than two on the scales below 300 nm and approaching three on the larger scales.

Bancaud and coworkers [12] studied the diffusive behavior of tracers in living mouse cells. Owing to crowding-induced enhanced affinity, chromatin-interacting proteins remained transiently trapped in heterochromatin. The dynamics of soluble nuclear proteins were demonstrated to be independent of their size, which gave rise to a fractal model of chromatin organization. The fractal architecture differed between heterochromatin and euchromatin, with a higher fractal dimension for euchromatin, a rougher and more branched surface, which filled more of the 3D space of the nucleus. The lower fractal dimension of heterochromatin indicated a smoother outline, with a smaller surface area.

Polymers can reveal fractal characteristics [13]. Fractal structures can be created in polymers by iterative processes. During condensation, a polymer is repeatedly subjected to the self-similar process of crumpling. Thus, a repeatedly folded, condensed molecule finally becomes a fractal. In this way, a long polymer can be packed in a small volume without crossing itself and without entanglements. This elegant form of packaging facilitates rapid unraveling when necessary and is therefore advantageous [14,15]. A recent experiment suggests that this process also applies to chromatin and can, in a simplified manner, illustrate its fractal properties. With the help of the Hi-C technique, Lieberman-Aiden et al.[14] simultaneously analyzed adjacent DNA across the entire genome of K562 and GM06990 cells. They identified a genome organization characterized by the spatial segregation of open and closed chromatin, building two genome-wide compartments. The chromatin conformation can be described by their model as the fractal globule. This is a knot-free polymer conformation that enables maximally dense packing but maintains the ability to easily fold and unfold any genomic locus. The new model of the fractal globule substitutes the former paradigm of the globular equilibrium model.

All these experiments favor the concept of the fractal nature of DNA, nuclear chromatin and the surrounding nucleoplasmic space. In other words, today there is strong evidence for a fractal organization of the nucleus. Yet, morphologists (histologists, cytologists and pathologists) demonstrated indirect evidence for the fractal organization of chromatin for nearly two decades [16–29].

If a 3D fractal structure, for example, the chromatin arrangement in the nucleus, is cut in planes, the resulting bidimensional (histological or ultratsructural) images also reveal fractal characteristics. This happens when we make tissue sections of a nucleus. Even applying different stainings, with the help of computerized image analysis, the fractal nature of chromatin was demonstrated in histologic and ultrastructural sections, as well as in cytologic smears, which reveal a bidimensionally spread nucleus.

Morphologists differentiate two distinct chromatin conformations: first, the uncondensed euchromatin and the much denser and darker heterochromatin, which is usually considered to be transcriptionally less active. Structure and function interact mutually. Changes of the nuclear architecture, as seen in histological or cytological preparations, reflect genomic and nongenomic changes, which are very common in tumor cells. Second, late-onset translocations and gene mutations have been implicated in the disease progression of malignant neoplasias. Neoplastic cells often reveal increased gene expression with alterations of chromosomal positioning. Therefore, it was hypothesized that relocation of the chromosomes may provoke alterations of the chromatin pattern in malignant cells [30].

Furthermore, epigenetic modifications play a major role in the pathogenesis and progression of tumors [31,32]. Malignant tumors show widespread epigenetic changes, including global hypomethylation, especially of repeat DNA sequences, as well as focal hypermethylation of multiple CpG island gene-regulatory regions. Hypomethylation of repeat sequences is associated with decondensation of the chromatin structure. It induces chromosomal instability and is significantly associated with tumor progression [31–33]. Strong hypomethylation is associated with poor prognosis [31]. Furthermore, malignant growth may influence histone modifications, which are epigenetic regulators of chromatin and are important for the compartmentalization of the genome [32,34]. DNA methylation and histone modifications often provoke concomitantly aberrant gene expression in many tumors [34].

Recently, it was hypothesized that malignant neoplasias could not only reveal genetic but perhaps also epigenetic instability [35]. A more aggressive behaviour is usually found in genetically unstable neoplasias with a considerable (and, during tumor progression, increasing) number of genetic or epigenetic changes. Therefore, unstable tumors with a poor prognosis should reveal a more complex chromatin rearrangement, with a mixture of many chromatin areas with varying density (lighter and darker). This would be equivalent to a higher fractal dimension in the computerized image analysis [29].

Indeed, an increased fractal dimension of chromatin was an independent adverse prognostic factor for survival of patients with different malignant neoplasias, such as multiple myeloma [28], squamous cell carcinoma of the oral cavity [24], squamous cell carcinoma of the larynx [20] and malignant melanoma of the skin [29]. This implies that the complexity of the chromatin architecture in cancer cells may reveal important prognostic information that is independent of other prognostic factors, such as staging, grading or even relevant cytogenetic aberrations.

The goodness of fit of the fractal dimension of chromatin in B-precursor acute lymphoid leukemia has been demonstrated to be of independent prognostic relevance, but as a favorable prognostic factor. In other words, patients with a chromatin architecture more similar to the ‘ideal mathematical fractal‘ had a better outcome [2].

Explanations for this finding are speculative at present. However, it might be possible that the great number of genetic and epigenetic modifications in aggressive tumor clones could disturb the process of auto-organization in the nucleus to such an extent that the self similarity of the chromatin structures across different scales is not as perfect as in less aggressive clones.

In summary, both the fractal dimension and its goodness of fit permit the rough estimation of chromatin rearrangement complexity in a global manner in tumor cells. Several studies indicate that both features may reveal new and biologically relevant prognostic information. The methods used are simple, reproducible and inexpensive, and may be applied on routine slides from the archives, thus permitting retrospective studies without any additional costs.

Financial & competing interests disclosure

The author has received grants from the São Paulo Research Foundation FAPESP (project 2007/52015–0) and the National Council of Technological and Scientific Development CNPq (project 309447/2007–0). The author has no other relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed.

No writing assistance was utilized in the production of this manuscript.

Additional information

Funding

The author has received grants from the São Paulo Research Foundation FAPESP (project 2007/52015–0) and the National Council of Technological and Scientific Development CNPq (project 309447/2007–0). The author has no other relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed. No writing assistance was utilized in the production of this manuscript.