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Computer modeling of very large biomolecular systems, such as long DNA polyelectrolytes or protein-DNA complex-like chromatin cannot reach all-atom resolution in a foreseeable future and this necessitates the development of coarse-grained (CG) approximations. DNA is both highly charged and mechanically rigid semi-flexible polymer and adequate DNA modeling requires a correct description of both its structural stiffness and salt-dependent electrostatic forces. Here, we present a novel CG model of DNA that approximates the DNA polymer as a chain of 5-bead units. Each unit represents two DNA base pairs with one central bead for bases and pentose moieties and four others for phosphate groups. Charges, intra- and inter-molecular force field potentials for the CG DNA model were calculated using the inverse Monte Carlo method from all atom molecular dynamic (MD) simulations of 22 bp DNA oligonucleotides. The CG model was tested by performing dielectric continuum Langevin MD simulations of a 200 bp double helix DNA in solutions of monovalent salt with explicit ions. Excellent agreement with experimental data was obtained for the dependence of the DNA persistent length on salt concentration in the range 0.1–100 mM. The new CG DNA model is suitable for modeling various biomolecular systems with adequate description of electrostatic and mechanical properties.

During decades, the DNA molecule has been studied in a vast number of experimental, theoretical, and modeling research investigations. The study of polyelectrolyte properties of DNA takes an important place in these works since electrostatic interactions and salt effects have a profound influence on many other DNA properties and interactions of DNA with its surrounding. An important problem in this context is related to understanding the mechanism of DNA compaction and self-assembly. In the nucleus of eukaryotic cells, the DNA molecule exists as a hierarchically ordered structure formed by the nucleosome core particles consisting of complexes of DNA with histone proteins, and there are strong indications [

An accurate description of molecular properties of condensed matter can in principle be reached by using atomistic molecular dynamics computer simulations. With computer simulations, we are able to follow every detail of the molecular motion and give, within some theoretical model, quantitative answers, directly comparable with experiments, thus, providing a link between the theory and the experiment. This is especially important in studies of macromolecular systems, such as biopolymers, which always have a certain degree of intrinsic disorder. At present, computer simulations have reached a state where simulation times and length scales allow a direct comparison with experiments in well-characterized systems. Molecular simulations are also becoming an intrinsic part of applied research, such as drug design, materials science, and nanotechnologies.

In order to model very large condensed matter systems such as DNA-nucleosome complexes, an atomistic description is not practical as the number of particles would be very large. Therefore, simplified coarse-grained (CG) models are necessary, allowing a coarser description of the molecules and interactions when moving towards description at larger length and time scales. Many of today’s coarse-grained models are constructed from empirical parameterization of effective interaction potentials and usually do not contain a rigorous description of the molecular water [

A number of theoretical coarse-graining models of DNA [

We have previously performed modeling of an NCP solution [

An advanced coarse-grained DNA model was recently introduced by us [

Coarse-Grained DNA model and definition of sites and intramolecular interactions: (

In the present work we parameterize the coarse-grained DNA model introduced in [^{+} as counterions. The internal coarse-grained DNA potentials are then computed by the inverse Monte Carlo method using the MagiC software [_{P}) data was obtained for the salt-dependent bending behavior of DNA over a wide range of salt from 0.0001 M to 0.1 M, while the torsional rigidity of the present CG-DNA model requires further optimization. Potentially the proposed CG model could be further developed to include sequence-specific dependence of DNA flexibility. However, this CG model does not include modeling separation of the DNA strands.

The aim of the present CG approach is to develop a model that is simple enough in its design and yet captures the structural form and properties of double helical B-form DNA, which is suitable for rigorous parameterization based on all atom MD simulations followed by the inverse Monte Carlo method to obtain relevant CG DNA structural and interaction potentials. The model should also pay particular emphasis to electrostatic interactions, which includes explicit mobile ions and explicit charges of DNA phosphate groups and it should also have the potential for refinements, such as inclusion of base sequence specificity (presently not considered).

To satisfy this level of precision, we have chosen to describe the system via a collection of coarse-grained beads representing different fragments of the DNA with intramolecular bonded potentials that maintains the DNA structural integrity while reproducing experimentally observed flexibility of the B-DNA double helix. DNA is modeled as a sequence of units each representing a two base pair fragment of DNA. Each unit consists of five beads, four P units that represent the phosphate groups and one D unit corresponding to the four pentoses with attached base fragments in between (

Parameters of the coarse-grained (CG) DNA model developed from the inverse Monte Carlo analysis [

Type of bond/angle | Equilibrium |
IMC derived force constant_{B}T·Å^{−2} or k_{B}T·rad^{−2}) |
---|---|---|

D–D bond | 7.14 Å | 14.8 |

D–P bond | 9.04 Å | 9.7 |

P–P bond | 6.88 Å | 19.4 |

D–D–D angle | 166.00^{°} |
37.0 |

P–D–P angle | 119.50^{°} |
100.0 |

P–P–P angle | 165.00^{°} |
32.0 |

Atomistic MD simulations were run for four 22-bp DNA oligomers (sequence 5’-d(GATGCAGTCACCGCGAATTGGC) × 5’-d(GCCAATTCGCGGTGACTGCATC)) dissolved in 21,200 water molecules with 168 K^{+} counterions. The initial structure of the DNA double helix was built using the NAB package [^{−5}. Simulations were run for 40 ns collecting atom coordinates with 1 ps resolution. Detailed description of the simulations and major results are reported in our earlier work [

The trajectories of the atomistic simulations were mapped onto the CG DNA model producing a CG trajectory. The P sites were determined by the positions of the phosphates of the atomistic model and the D cites were computed as center-of-masses of the atoms belonging to the two base pairs involved. Superposition of the atomistic DNA structure on our 5-site CG-model is displayed in ^{+}), totally 15 pairs of intermolecular distribution functions were computed from the CG trajectory. Additionally, distributions of bond distances and angles, corresponding to bond and angle internal DNA potentials (see

We also performed inverse Monte Carlo calculations without distinguishing the oligomer terminal groups in order to check their effect. As expected, the effect of the terminal groups was non-negligible and we discuss this in the Results Section below.

In the present work we have used the results of the inverse MC procedure to obtain the parameters that determines the internal structure and flexibility of the coarse-grained DNA model, the topology of which is illustrated in

The bond and angle potentials for the bound sites were approximated by the equations:

where _{b}, _{a}, and _{0}, φ_{0} are respectively bond and angle force constants and equilibrium values for bond length and angle. The final parameters for the bonded interactions are given in

During this first test of the model, only internal DNA rigidity parameters were fitted to the results of the inverse Monte Carlo computations, while the standard Coulombic potential Equation (3), for charges (_{i}, _{j}) in a dielectric continuum with permittivity ϵ = 78, and short-range potential defined by Equation (4) were used to describe non-bonded interactions:

Here (i, j) are bead numbers. In Equation (4) the hard core particle contact distance _{ij} = _{i} + _{j} is determined by the values of the hard core radii of both beads. Parameters σ_{ij} and _{ij} = 4 Å and _{B}_{ij}/2 = 2 Å and the hard radius _{i} which was defined separately for each particle type. The hard radii _{i} as well as charges _{i} are given in

In the present CG application, which aims to parameterize the DNA internal parameters with the purpose of reproducing its salt dependent bending flexibility, we consider the solvent as a dielectric medium. These interaction potentials represent an approximation to the solvent-mediated effective potential between the charged particles in the solvent medium. Thus, ε is a parameter of the effective ion-ion potential. Many studies at an all atom level have shown that the effective potential of mean force of ions in water usually have one or two oscillations around the Coulomb potential with ϵ = 78 for small (within 8 Å) distances between the ions [

In a dielectric continuum model the ion interactions play an important role. It is important to assign appropriate radii, reflecting the effective hydration of the ions. Here, the following values of the hard radii were used: ^{+}) = ^{−}) = 0, which corresponds to effective radii of 2.0 Å for these monovalent ions. The phosphate group of DNA was approximated as a bead with radius ^{−}) = 1.0 Å, which gives an effective radius of 3.0 Å. The choice of sizes for K^{+} and Cl^{−} is justified by our previous extensive comparison of experimental as well as computer modeling results of ion-DNA, ion-NCP and ion-chromatin interactions and ion-ion competitions [

Non-bonded parameters of the coarse-grained DNA model.

Site | Charge | Radius _{i} (Å) |
---|---|---|

D (DNA) | +0.8 | 4.0 |

P (DNA) | −1.2 | 1.0 |

K^{+} (ion) |
+1.0 | 0.0 |

Cl^{−} (ion) |
−1.0 | 0.0 |

The CG DNA model defined by the parameters given in ^{8} MD steps in each simulation run where the first 5 × 10^{7} steps were considered as equilibration. The Langevin dynamics simulations were performed by the ESPResSo package [

We also confirmed that the local structure of DNA remains preserved in comparison to the all-atom simulations from which its structural parameters were derived. The distribution of distances for some pairs of DNA beads were computed (data not shown). The D–D distance showed a somewhat extended average value (7.3 Å, compared to 7.14 Å in the all-atom simulations), corresponding to an increase of about 0.1 Å per base pair. On the other hand, the distribution of P–P bond distances is the same as in the all-atom simulations (with an average of 6.9 Å). These average distances show that the DNA model is robust, while still allowing for fluctuations and that the main features of the double helical DNA are maintained and close to that of the B form structure.

Parameters of Langevin simulations (box size and the number of ions) of the coarse-grained DNA model. DNA length was 200 base pairs in all cases.

Ionic strength | Box size |
Number of cations^{+}) |
Number of anions^{−}) |
---|---|---|---|

Uncharged ^{1} |
800 | 300 | 300 |

100.0 mM | 500 | 7900 | 7500 |

30.0 mM | 500 | 2650 | 2250 |

10.0 mM | 800 | 3400 | 3000 |

3.0 mM | 800 | 1300 | 900 |

1.0 mM | 800 | 700 | 300 |

0.3 mM | 1400 | 900 | 500 |

0.1 mM | 1420 | 580 | 180 |

^{1} Note: charges of the D and P sites were set to zero.

The effective coarse-grained potentials were originally obtained in a tabulated form. In order to facilitate their use in general purpose molecular dynamics software, we approximated the internal bond and angle potentials of DNA by harmonic functions. The results of original potentials for non-terminal DNA sites (output from the inverse MC calculations), their harmonic approximation, as well as the distribution of the corresponding bond lengths and angles (calculated from the all atom MD simulations), are shown in

Merging terminal “P” and “D” particles “contaminates” the statistics, which uses only the central 9 (18 base pairs) DNA units and some of the potential functions lose their parabolic shape. RDFs and potentials for this simplified coarse-graining are given in

Radial distribution (RDF, solid lines) and force potential (dashed curves) functions for the internal bonds (

In addition to the internal bond and angle distributions of the DNA molecules, the combination of the all-atom MD simulations and the inverse MC method generated a set of intermolecular distributions and potentials. This data is presented in

To validate the CG-model of DNA we carried out a series of simulations for a 200 bp fragment of DNA in solution with varying concentration of monovalent salt. The purpose of the simulations was to determine the dependence of the mechanical properties of the CG-DNA, mainly bending on salt concentration and to compare these properties with experimental data. Details of the simulations are given above (_{P}, was determined from the well-known relationship [

where 〈cosα〉 is the average of the cosine of the angle between two neighbouring segments over the simulation and _{C} is the contour length between the segments. There is a technical problem preventing a straightforward calculation of the contour length and assignment of a vector to a DNA segment of the present CG-model of the DNA. The positions of the “D” beads (defined as the center of mass (COM) of the pentose and base atoms of the two DNA base pairs) are not aligned along the axis of the double stranded DNA and form a thin helical structure (see

For an ideal worm-like chain Equation (6) is exact for a given value of _{P}, but for a real DNA model the _{P} determined by Equation (6) depends on the chosen contour length _{C} as well as on the precise definition of the segment. In practice, the _{P} value was calculated from the slope of the initial 20 points of the dependence _{C} _{C} values as for the full set of data (data not shown).

Inspection of snapshots showed that the DNA molecules had a visible difference in degree of bending when comparing the system with different influence of electrostatic forces. The DNA with charged P and D particles and in low 0.1 mM salt (_{P} calculations for a range of salt concentrations. Our simulation data showed a remarkably good agreement with experimental _{P} values (

Determination of the persistence length, _{P}, from the slope of dependence of DNA contour length, _{C}, on In(〈cosα〉). Data calculated for different concentrations of monovalent salt and for the uncharged CG-DNA are displayed as points; thin red lines show linear fitting.

Dependence of DNA persistence length, _{P}, on monovalent salt concentration. (_{P} value calculated for the CG DNA model with charges on the phosphate group beads set to zero. Experimental data are taken from: black circles [

Flexibility of the CG-DNA as a function of concentration of monovalent salt.

Cation concentration ^{1}, |
Persistent length, _{C} |
Torsional persistence length, _{T} |
---|---|---|

0.104–0.340 | 1065 ± 35 | 1446 |

0.303–0.545 | 912 ± 20 | 1461 |

0.970–2.270 | 803 ± 17 | 1415 |

2.910–4.210 | 633 ± 7 | 1463 |

9.730–11.030 | 606 ± 10 | 1409 |

29.800–35.200 | 431 ± 4 | 1424 |

99.600–105.000 | 451 ± 3 | 1399 |

No charge | 316 ± 1 | 1349 |

^{1} Note: it is incorrect to define the NaCl concentration (_{NaCl}) alone as the parameter characterizing the ionic environment since the dissociated DNA counterions contribute to the ion concentration in the simulation cell. Therefore a range of concentrations are used where lower value is _{NaCl} and the higher one is _{NaCl} + _{P} (where _{P} is total concentration of DNA counterions).

Recently, the dependence of DNA elastic properties on ionic environment and analysis of the relative contributions of electrostatic (due to phosphate-phosphate repulsion) and structural (mostly due to base-base stacking) factors to the DNA stiffness has been investigated using all-atom MD simulations followed by so-called group renormalization calculation of effective potentials used in a CG model of DNA [

The torsion persistence length is determined from the following procedure. A torsion angle is defined between all neighboring monomers using vectors from the central bead position to COMs of adjacent phosphates. A torsion angle between more separated monomers (like _{C}) is determined as the sum of torsion angles between all intermediate neighbors. Then the torsional persistence length is defined from [

where δφ^{2} = 〈φ^{2}〉 ‒ 〈φ〉^{2} is variation of the torsion angle φ between monomers separated by _{C} distance.

In all MD simulations of the 22 bp CG DNA, the dependence _{C} ^{2} shows excellent linearity for full range of _{C} (Fig S3A of the Supporting Material) giving values of _{T} in the range 1350–1500 Å (_{T} values around 1400 Å thus show that the torsional resistance in the CG model is exaggerated. However, the salt-dependence displays minor effect on the torsional rigidity of DNA (

We present a CG DNA model that consist of two types of beads representing a set of two base pairs, one P bead for each phosphate group and one central D bead for the pentose and bases of a two-bp unit (_{P}. The CG model combined with the parameters obtained directly from the all-atom simulation without further adjustment or optimization, resulted in excellent agreement for the dependence of _{P} on monovalent salt concentration when compared with available experimental data. The results are in agreement with a recently developed two-bead (per bp) CG DNA model [

We also made preliminary analysis of the torsional persistence length of our CG DNA model. While the model produce the expected independence of this parameter on salt, the absolute values for torsional persistence length seem too large and will require further optimization in future work.

The conclusions from our analysis regarding the relative contributions of the electrostatic and elastic forces that defines the bending properties of DNA under physiological conditions is also in agreement with the work of Papoian, Savelyev, and co-workers [

This work has been supported by the Singapore Agency for Science Technology and Research (A*STAR) through the Biomedical Research Council (LN), by the Singapore Ministry of Education Academic Research Fund (AcRF) through a Tier 1 grant (LN) and by the Swedish Research Council (to APL).

N.K., A.P.L. and L.N. designed and planned simulations and analysis and wrote the paper; N.K., D.L. and A.L. performed simulations and analysis of data.

The authors declare no conflict of interest.

^{+}, Na

^{+}, and K

^{+}ions to DNA in oriented DNA fibers

^{2+}, Ca

^{2+}, Na

^{+}, and K

^{+}to DNA in oriented DNA fibers: Experimental and Monte Carlo simulation results

^{+}and K

^{+}ions in oriented DNA fibers

Supporting Material (PDF, 176 KB)