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**3. Approach**

In the late 1950's, Eugene Wigner showed how quantum mechanics can
limit both the accuracy and precision with which a clock can measure distances
between events in space-time [6]. Wigner's clock inequalities can be written as
constraints on the accuracy (maximum running time Tma%) and precision (smallest
time interval TmiB) achievable by a quantum clock as a function of its mass M ,
uncertainity in position X , and Planck's constant h :

Heisenberg uncertainty principle, which requires that only one
single simultaneous measurement of both energy (E = mc2) and the time 7"min
be accurate. Wigner's constraints require that repeated measurements not
disrupt the clock and that it must accurately register time over the total
running period Tmm . An intuitive way of saying this is that a Wigner clock
must have a

minimum mass so that its interaction with a quantum of light
(during the measurement of a space-time interval) not significantly perturb the
clock itself. Wigner suggested that these inequalities (Eqns. 1 and 2) should
fundamentally limit the performance of any clock or information processing
device [7], even "when the most liberal definition of a clock" is
applied [6].These inequalities have elegantly been applied by John Barrow [7]
to describe the quantum constraints on a black hole (a rather "liberal
definition for a clock"). He shows that the maximum running time for a
black hole (rmax in Wigner's relations) corresponds to its Hawking lifetime and
that Wigner's size constraints are equivalent to the black hole's Schwarzchild
radius. Furthermore, Barrow demonstrates that the information processing power
of a black hole is equivalent to its emitted Hawking radiation. Wigner
inequalities should likewise provide nontrivial constraints on the performance
of any information processing nanomachine or time-registering device.Here we
heuristically examine the ramifications of these limits on the capability of a
nanomachine to read DNA. We assume that A is the uncertainty in the motor's
position along the DNA (linear span over which it processes information) and
can be estimated by the length of the DNA molecule (e.g. ~ 16//m for
lambda-phage DNA used in typical single molecule experiments).

Then Eqn 1.1 gives rmax < 387 sec as the maximum running time
for which the motor can reliably run and still be accurate. For comparison, the
error rate for a polymerase motor from the species Thermus aquaticus (TAQ) is
about 1 error every 100 sec. Likewise,
Eqn 1.2 gives rmjn > 5x10 14 sec as the motor's
precision, or the minimum time interval that it can measure. A plausible
interpretation of this value is that Tmin corresponds not to the motor
incorporating one base but to the motor undergoing an internal state
transition. Note this time corresponds well to the timescale for the lifetime
of a transition state [8].These Wigner constraints can also be written in terms
of Imax, the maximum readout length over which the motor is accurate, and Lmm,
the minimum effective step size of the motor. If the motor's speed VmMr ~ 100
bases/sec, then Lma!. = Vmtu x rmax ~ 4xio4bases. This compares reasonably well
with known error rates of the DNA polymerase motor. For example, a TAQ
polymerase is known to make a mistake once for every 104 bases it reads.
Likewise, Imin = Vmom x rmin ~ 5xio l2 bases, which corresponds to about 2xio21
m. This is the effective step size or the minimum "distance" interval
that can be accurately registered by the motor. This linear coordinate
corresponds to the time associated with the fastest internal state transition
within the motor and indicates a minimum lengthscale over which the motor can
register information.