Model of Fractal Organization of Chromatin in Two-Dimensional Space

Chromatin, consisting of a meter-long DNA strand and associated proteins, is packed into the nucleus of a biological cell tightly but without entanglement. There is a hypothesis, confirmed by experiments involving the chromatin conformation capture technology [1], that curves densely filling the space (Peano or Hilbert curves) provide a good theoretical model to describe the chromatin packing into the nucleus. However, small-angle neutron scattering (SANS) experiments show a bifractal organization of chromatin in the interphase nucleus, thus demonstrating the presence of a logarithmic fractal on larger scales and a volume fractal on smaller scales [2]. In this paper, numerical Fourier analysis in the two-dimensional space is applied to simulate neutron scattering, and a model of a unified bifractal object is presented. It is shown that, in numerical radiation scattering experiments in the two-dimensional space, the mass and logarithmic fractals are significantly different from space-filling curves and from nonfractal objects. For instance, for a logarithmic fractal with a Hausdorff dimension of 2, scattering intensity decreases with increasing Fourier coordinate q by the power law q–2. For curves filling the two-dimensional space, the intensity decreases by the power law q–3, just as for nonfractal objects with sharp boundary in the plane. Thus, first, it is demonstrated that the model of space-filling curves is inadequate to describe the chromatin packing into the nucleus of a biological cell; second, a model of a unified bifractal object is proposed that combines logarithmic and mass fractals on different scales; and, third, a model of chromatin packing is proposed that can describe the data of both small-angle neutron scattering experiments and experiments involving chromatin conformation capture technology.