Single molecule measurements and biological motors

Thermal motion of trapped objects

Forces in an optical trap.
a) Schematic of a 1 μm bead diffusing in an optical trap. The bead's position is monitored with a detector, shown schematically above the trap. The forces acting on the bead are the restoring force due to the optical trap, stiffness κ; the viscous drag force on the bead, drag coefficient γ; and thermal forces due to the mean thermal energy (kT/2).
b) Mechanical model of the forces acting on the bead. The spring represents the restoring force due to the trap stiffness; the dashpot the damping due to viscous drag; and the arrows the random forces due to thermal energy.

A micrometer sized bead in water has an extremely low Reynolds number, which means that inertial forces play a negligible role in the bead's motion. The mass of such beads is tiny and their density often similar to that of water, so that, over short timescales at least, we can also disregard gravitational force. This means we need only to consider the following forces acting on a trapped object (see schematic figure above):

Restoring force due to the optical trap: F = κ·x where x is the displacement from the centre of the trap and κ is the trap stiffness.

Drag force due to the viscosity of the medium: F = γ(dx/dt) where γ is the drag coefficient and dx/dt is the velocity.

Thermal forces (F_t) due to thermal energy (kT). This force varies randomly and results in the so called thermal or Brownian motion of the object.

The forces acting on a freely diffusing but trapped particle at any moment can then be described by the equation:

F_t = κ·x + γ(dx/dt)

Where we are using optical tweezers to make measurements on motor protein or other molecules, we must also consider the stiffness of, and forces exerted by, these molecules. Each of these terms, and their implications, is discussed in more detail below. See also Svoboda and Block (1994), and Howard (2001). The figure below shows an actual recording of the random motion of a trapped object (a latex sphere).

Thermal motion of a trapped object. A 1 μm diameter polystyrene sphere was held in a weak optical trap and its position recorded with a quadrant photodiode at 5 kHz. Note the random nature of the motion.

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