Constant-Force Pulling on a Lennard-Jones Polymer

Patrick O’Neill

CHE210D Spring 2009 Final Project

Summary

                The end-to-end distance of a linear Lennard-Jones polymer is examined, with one end fixed to a surface, and a constant pulling force applied to the other end. The polymer is initially in a coiled state, and uncoils with sufficient pulling force. The average uncoiling rate depends on the pulling force, and for insufficient pulling force, the polymer remains in the coiled state for long times.

Background

                Atomic force microscopy and optical tweezers have been used to characterize the force-induced mechanical properties of a wide range of proteins. The simple model presented here is motivated by such experiments, in which a polymer is fixed at one end and pulled at the other to induce an unfolding transition.

Simulation methods

                The polymer is modeled as a linear chain of atoms, with adjacent atoms bonded by a harmonic potential, and non-adjacent atoms interacting through a Lennard-Jones potential. One end of the polymer is fixed to the y = 0 plane and all the other atoms feel a short-ranged, soft repulsion from the surface.

                Molecular Dynamics simulations are performed using a simple solvent model, Langevin’s thermostat, with force equations for atom q given by:

m d²x_q/dt² = F_q,x(r₁, ... , r_N) - mγ dx_q/dt + ξ_x(t) + F_pull,           etc. for y, z

The first term includes contributions from a harmonic potential between adjacent atoms, a Lennard-Jones potential between non-adjacent atoms, and a soft repulsion between the atoms and the y = 0 surface. The second and third terms account for the viscous and randomizing effects of the solvent. The last term is a constant pulling term applied to the free end of the polymer.

                The velocity Verlet algorithm is used to integrate the equations of motion, with a time step Δt. Random forces are calculated at each time step according to:

ξ_x(t) = η_x(t) ∙ Ö(6mgk_bT/Δt)                                                                                                                              etc. for ξ_y(t), ξ_z(t)

where the random numbers η_x(t), η_y(t), η_z(t) are independent Gaussian variables of vanishing mean and unit variance. Results here are presented for m = 1, γ = 10, k_bT = 1, Δt = 0.0001,

σ_LJ = 1, ε_LJ = 100.

                An initially extended polymer is allowed to coil into a compact state under zero pulling force. This coiled polymer is then pulled with a constant force, and the end-to-end distance is monitored as a function of time. Pulling forces are chosen empirically to cover the range of phenomena  from fast-unfolding to no-unfolding.

Figure 1: Constant-force extension curves for F_pull = 200, 400, 600, and 100.

Figure 2: Constant-force extension curves for F_pull = 400. The average over 10 trials is shown in black. A typical time trace is shown in blue, and a more dramatic uncoiling event is shown in red.

Results and interpretation

                For sufficiently large pulling forces, the end-to-end distance of the polymer grows approximately linear over time until it reaches ~90% of its full length, where it starts to level off. This trend can be seen on different time-scales for different pulling forces, for example F_pull = 1000 in Fig.1 and F_pull=400 in Fig.2. When the pulling force is too small, the end-to-end distance does not grow over time. For example, with F_pull = 200, the mean end-to-end distance at T = 1000 is very similar to that at T = 0 (not shown).

                These initial results indicate that the simple Lennard-Jones polymer pulling experiments might be useful for very simple studies of polymer unfolding under external force. It would be interesting to extend these simulations, for example by reducing the Lennard-Jones energy scale to see how much work goes into “pulling against potential energy” vs “pulling against backbone confirmational entropy”. It may be necessary to ramp the pulling force for such studies.

                The Lennard-Jones polymer model does not account for the specific interactions between protein side groups. It has just enough details to allow for coiled and extended states, but sequence specific models are necessary to model real proteins. Furthermore, proteins may bind to surfaces at multiple locations requiring more detailed models of the protein-surface interaction. At least one online tutorial already exists for constant speed pulling, using Go-type protein models.

Movie

LJ_polymer_pulling.avi

LJ_polymer_pulling.mpg

The movie shows the uncoiling of a 30-atom Lennard-Jones polymer with F_pull = 600. The the polymer starts to uncoil from the free end (pulled end), with some coiled region remaining near the fixed end. Gradually the coiled region becomes smaller and smaller as the polymer is stretched into a completely extended state.

Source code

source.zip

References

The form of random force used in the Langevin dynamics simulations is from:

“Molecular Dynamics Simulations of the Lennard-Jones Polymers in a Good Solvent”

Ciesla, M. et al. Acta Physica Polinica B, 38, 1727-1736, (2007)

An online tutorial for constant speed pulling of proteins using a Go-type model can be found at:

http://mmtsb.org/workshops/mmtsb-ctbp_2006/Tutorials/Go_Pulling/GoModelPullingTutorial.html