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.