**3.2 Estimation of Decoherence Times of the Motor-DNA Complex**

We consider the motor-DNA complex as a quantum system moving in
one dimension where the environment is a heat bath comprised of a set of n
harmonic oscillators, each vibrating with a given frequency and a given
coupling strength between oscillations. Then, using an expression derived by
Zurek [10], we can estimate the decoherence time of this system as where the
motor of mass M is in a superposition of two position states that are separated
spatially by AJC . This effective interstate spacing Ax can be estimated as
L/Mmea— 10"15 m, for a motor that takes roughly 1010 computational steps
while reading a 16 micron long molecule of DNA. The thermal de Broglie
wavelength Xj for a motor in equilibrium with its
surrounding heat bath is estimated

When the thermal de Broglie wavelength is much smaller than the
distance Ax , the motor-DNA system will behave as a classical system. On the
other hand, when the thermal de Broglie wavelength is on the order of, or
larger than the spacing Ax, quantum effects will predominate. Thus, for our
motor-DNA complex, Eqn. 1.6 gives tD ~8.4xl04 tr. The spectrum of relaxation
times for a DNA polymer vary with the length of the molecule and can be
estimated from the Zimm model [11] and have been experimentally verified [12]
to range from microseconds to milliseconds. For instance, the slowest
relaxation time for a DNA polymer chain of length L and persistence length P
can be approximated via the Zimm model.

This corresponds to about the longest relaxation time being about
500 milliseconds for double-stranded DNA and about 3 milliseconds for single
stranded DNA. With the longest DNA relaxation times being in the milliseconds,
the corresponding longest decoherence times (Eqn 1.8) of the motor-DNA complex
will range from several minutes to several hours. This easily satisfies the
condition that tD » Tbasemdi„x ¦ Thus, it is indeed quite possible that quantum
mechanical effects play a proactive role in influencing the dynamics of motors
reading DNA.