An associative memory Hamiltonian model for DNA and nucleosomes

Abstract

A model for DNA and nucleosomes is introduced with the goal of studying chromosomes from a single base level all the way to higher-order chromatin structures. This model, dubbed the Widely Editable Chromatin Model (WEChroM), reproduces the complex mechanics of the double helix including its bending persistence length and twisting persistence length, and the temperature dependence of the former. The WEChroM Hamiltonian is composed of chain connectivity, steric interactions, and associative memory terms representing all remaining interactions leading to the structure, dynamics, and mechanical characteristics of the B-DNA. Several applications of this model are discussed to demonstrate its applicability. WEChroM is used to investigate the behavior of circular DNA in the presence of positive and negative supercoiling. We show that it recapitulates the formation of plectonemes and of structural defects that relax mechanical stress. The model spontaneously manifests an asymmetric behavior with respect to positive or negative supercoiling, similar to what was previously observed in experiments. Additionally, we show that the associative memory Hamiltonian is also capable of reproducing the free energy of partial DNA unwrapping from nucleosomes. WEChroM is designed to emulate the continuously variable mechanical properties of the 10nm fiber and, by virtue of its simplicity, is ready to be scaled up to molecular systems large enough to investigate the structural ensembles of genes. WEChroM is implemented in the OpenMM simulation toolkits and is freely available for public use.

Fig 1. An Associative Memory Hamiltonian for DNA.

(a) Comparison of an atomistic representation and a coarse-grained representation of a B-DNA dodecamer (PDB ID: 1BNA [55]). (b) The AMH recapitulates the energy landscape of the DNA molecule using a series of Gaussian wells. The parameters $r_{ij}^m$ are determined using the distances measured in the structural memories, while the energy of interactions, λ and the scale of structural fluctuations, σ, are tuned to reproduce experimental observables. (c) Schematic representation of the energy terms in the model: connectivity (springs), excluded-volume terms (dashed circle), and AMH terms encoding the double helix structure (red dashed lines). An AMH term is applied between any pair of particles inside the box of 4bp. The box is then slid along the DNA, and the procedure is repeated.

Fig 2. DNA Persistence Length and Parametrization.

(a)(b) These figures show how persistence lengths depend on the strength of the energy interactions, λ, and the scale of structural fluctuations, σ. The bending persistence length is shown in panel (a), while the twisting persistence length is shown in panel (b). The σ is in reduced units (r.u.) which is 0.241 nm as explained in the Appendix A in S1 Document. Dots indicate the data points obtained by simulation; colored surfaces indicate their linear interpolation. The horizontal gray plane in (a) shows the experimentally determined bending persistence length of DNA, 150bp. This persistence length is set as the target for the optimization of the parameters. The red line indicates the intersection of the two surfaces, where our model meets the target persistence length value. Similarly, the gray box in (b) is the experimentally determined twisting persistence length, 75-360bp, our target in this case. (c) The figure shows the temperature dependence of bending persistence length. The four colored lines show results obtained using the WEChroM model. Black lines show experimental results published by Gray et al. [61] and Geggier et al. [62], respectively. As shown, the proposed model agrees well with the experimental observations for several couples of parameters (λ, σ).

Fig 3. Supercoiling behavior of DNA minicircles.

(a) Commonly observed shapes of DNA minicircles (b) Writhe analysis of DNA topoisomers -4 to 4. Positively supercoiled minicircles relax torsional stress forming plectonemes, which manifest in increasing writhe and are visible in the DNA conformations in (a). Moderately negatively supercoiled minicircles also form plectonemes, this time resulting in negative writhe. Strongly negatively supercoiled minicircles do not appear to form plectonemes. (c) Negatively supercoiled DNA relaxes torsional strain through local melting of the DNA double helix. The figure shows an example structure of a topoisomer -2, defects are indicated by the black and red arrows. (d) Bending angle along DNA for the minicircle structure in (c). Local defects are clearly visible at index 304 and 193, once again indicated by the black and red arrows.

Fig 4. Nucleosome model and unwrapping analysis.

(a) A detailed representation of the nucleosome core particle (PDB ID: 1KX5 [59]) (left) and the corresponding coarse-grained WEChroM representation (right). In the left figure, the upper half of the nucleosome particle is shown as solid while the lower half is transparent for clear visualization. The DNA nucleotides in close contact with amino acids are defined as “contact DNA” and subject to the AMH potential modeling histones-to-DNA interactions (black circle). In our model, contact DNA interacts with the histone octamer’s center of mass (U_center, represented by the red arrow in the figure). Contact DNA also interacts with neighboring contact DNA (U_neighbor, represented by the black arrows inside the indigo dashed circle). (b) Free energy of partial DNA unwrapping for a system composed of the histone octamer plus 223-bp of DNA; WEChroM model (blue) and 3SPN-AICG model (orange). Simulation data shown for the 3SPN-AICG mode are extracted from ref. [38], figure 8. Examples of nucleosome configurations are shown for end-to-end distances of roughly 45 Å (left), 107 Å (right), 330 Å (outside) (c) Free energy of partial nucleosome unwrapping for a system composed of the histone octamer plus 147-bp of DNA; WEChroM model. (d) Nucleosome AMH energy of the partial nucleosome unwrapping for the 147-bp WEChroM model. Example nucleosome configurations are shown for end-to-end distances of 76 Å (left), 142 Å (middle), 221 Å (right).