Entropic Elasticity of Double-Strand DNA Subject to Simple Spatial Constraints

The aim of the present paper is the study of the entropic elasticity of the dsDNA molecule, having a cristallographic length L of the order of 10 to 30 persistence lengths A, when it is subject to spatial obstructions. We have not tried to obtain the single molecule partition function by solving a Schödringer-like equation. We prefer to stay within a discretized version of the WLC model with an added one-monomer potential, simulating the spatial constraints. We derived directly from the discretized Boltzmann formula the transfer matrix connecting the partition functions relative to adjacent “effective monomers”. We have plugged adequate Dirac δ-functions in the functional integral to ensure that the monomer coordinate and the tangent vector are independent variables. The partition function is, then, given by an iterative process which is both numerically efficient and physically transparent. As a test of our discretized approach, we have studied two configurations involving a dsDNA molecule confined between a pair of parallel plates. One molecule end is anchored to one plate by a biochemical bond. A stretching force F , normal to the plates, is pulling away the other end. In the first case, the cristallographic length L is smaller than the two-plate distance L0. The molecule feels, then, only the anchoring barrier effect. The predicted elongation-versus-force curve, is pushed upward with respect to the WLC model result. This effect is the most spectacular in the low force regime. For large forces say, Fhigh = 5 kB T/A, the elongation versus L is very well fitted by a straight line with a slope given by the standard WLC model and a constant term ≃ 1.2A. In the second case, L takes values up to Lmax = 1.5L0. With a stretching force still equal to Fhigh, the standard WLC model predicts that the molecule cannot fit within the plates when L > L∗ = 1.29L0. We have studied the evolution of the elongation derivative with respect to L, together with the mean square free-end fluctuations along the force. They both exibit a sharp decrease when L ≥ L0. We present a semi-qantitative argument suggesting that the terminal segment involving 20% of the internal monomers flattens against the repulsive barrier when L → Lmax. In conclusion, we suggest extensions of the present work, relevant to the analysis of micromanipulation experiments. Finally, we have gathered into the Appendix formal developments, leading to a precise relation between the transfer matrix and the Hamiltonian methods for the study of spatially constrained dsDNA.