Models of macromolecular chains based on Classical and Quantum Mechanics: comparison with Gaussian models

Abstract

of contentsFeature Article: Different approaches to the study of open linear freely-jointed (unhindered) threedimensional macromolecular chains at thermal equilibrium, based upon either Classical Hamiltonian Dynamics with constraints (namely, a formulation equivalent to Kramers' one) or Quantum Mechanics (with stiff harmonic springs), are reviewed and discussed in detail. Results from those approaches are compared to those arising from the standard Gaussian model. Fraenkel's approach, based upon Classical Mechanics (with stiff harmonic springs) and its comparison with the Gaussian model are also treated. It is argued that the quantum formulation is required in order that the treatment of the high-frequency vibrations be consistent. The role of the uncertainty principle is discussed, in support of the quantum-mechanical approach. At a later stage, internal rotations can be treated through Classical Statistical mechanics, under certain conditions. The quantum-Mechanics-based model appears to be consistent with the Gaussian model, while the one based upon constrained Hamiltonian Dynamics yields certain discrepancies with the latter. Classical and quantum-mechanical approaches to open linear freely-rotating (hindered) threedimensional macromolecular chains are also treated: the consistency of the quantum-mechanics-based model is displayed by deriving approximately the existence of a persistence length. It is argued that the possible advantages of the quantum-mechanical formulation should be expected not at large scales but at shorter ones. The specific topics to be treated are described in the summary of contents given below.Macromol. Theory Simul. 9, No. 2

Flexible macromolecular chains with constraints: a quantum mechanical approach

“…The computations, which generalize non-trivially those in [15], are very lengthy and will be omitted. For previous (and far simpler) variational calculations, see [13] and references therein. One finds…”

“…, N −1, in HC,ang,res . Previously, a rather different variational computation for freely rotating macromolecular chains [13] has provided a different total zero-point energy (E 0 , dependent on all (u y j u y j +1 ) (0) ), restricted effective quantum angular Hamiltonian ( H ang,res ) and quantum partition function Z ang,res . A new result here is that, even if E 0 = E 0 , the above HC,ang,res and Z C,ang,res coincide with the classical limits H C,ang,res and Z C,ang,res of H ang,res and Z ang,res , respectively.…”

“…. , N. Equation (28) (which may be useful in numerical computations) implies, on the basis of [13] and some longdistance approximations, the existence of a persistence length d pl (>d). So, on a suitably large length scale, the classical freely rotating chain described by equation ( 27) can be approximated by a freely jointed one having N pl bonds (N pl < N − 1) and a new bond length d pl , with overall rotational invariance.…”

“…Moreover, they can be represented approximately by ground state wavefunctions and decouple consistently, yielding constant bond lengths and, eventually, bond angles and other angular constraints. See [12,13] for the above philosophy for freely jointed macromolecules (without angular constraints) and their consistency with the standard Gaussian model. Here, we shall generalize those ideas when angular constraints exist in flexible macromolecules and obtain reliable effective Hamiltonians and partition functions for them, depending only on the relevant unconstrained (slowly varying) angular degrees of freedom.…”

On a path integral description of the dynamics of an inextensible chain and its connection to constrained stochastic dynamics

“…Later on, mainly the statistical mechanics of such chains has been investigated, see e.g. [4][5][6]. Dynamical models are however interesting by themselves and have also some applications, for instance in modeling the response of a chain to mechanical stresses in micromanipulations [7].…”