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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 , 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  and have been experimentally verified  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.