**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**4.2.1 Broadband Receptor
Arrays**

One simple broadband concentration sensor, shown schematically^{10}
in Figure
4.1, consists of a graduated series of receptors having uniformly high specificity
but engineered with progressively greater affinities (successively smaller equilibrium
dissociation constants K_{d} (molecules/nm^{3}); Section
3.5.2) for the target molecule, which is present in the test sample at concentration
c_{ligand}. Exposure of the receptor array to the test sample for a
time t_{EQ} necessary to reach diffusion-driven equilibrium gives a
probability of receptor occupancy P_{occ} ~ 0.91 in a receptor with
K_{d} ~ 0.1c_{ligand}, P_{occ} ~ 0.50 in a receptor
with K_{d} ~ c_{ligand}, and P_{occ} ~ 0.09 in a receptor
with K_{d} ~ 10c_{ligand}.

During each measurement cycle, all steric probes are simultaneously
extended into their associated receptor volumes at a time t_{EQ} after
presentation of sample to the array. Probes which reach full extension register
an empty receptor; those which cannot fully extend register an occupied receptor.
After registration, the probes are retracted and ejection rods are thrust into
all receptors (typically requiring ~1 nanonewton (nN) of force for occupied
receptors) to empty them while the test chamber is flushed clear in preparation
for the next cycle to begin. Ideally, the probe rod pushing into the binding
site should push the bound molecule further into the binding site to avoid complications
that may arise due to competing kinetic barriers, and ejection rods should push
molecules far enough to ensure that the next resample is independent.

Consider a series of N steric probe units as proposed by Drexler.^{10}
Each probe measures 8 nm x (2.5 nm)^{2} ~ 50 nm^{3} and has
mass ~ 2 x 10^{-22} kg. The ratio of the dissociation constants of adjacent
units is k = K_{di} / K_{di+1} >
1, with K_{d1} ~ 10^{-4} molecules/nm^{3} and K_{dN}
~10^{-13} molecules/nm^{3} typically in the human body, and
so N = 1 +{log_{10} (K_{d1}/K_{dN}) / log_{10}
(k)}.

Minimum measurement error occurs when c_{ligand} exactly
matches the K_{d} of a probe unit, that is, when c_{ligand}
= K_{di} (i.e., P_{occupiedi} = 0.5). Maximum measurement error
occurs when c_{ligand} lies exactly midway between two probes such that
K_{di} > c_{ligand} > K_{di+1}, or, more specifically,
at the geometric midpoint c_{ligand} = (K_{di} K_{di+1})^{1/2}
= K_{di} k^{-1/2}. In this case,
Eqn. 3.24 becomes

where P_{unoccupiedi} = 1 - P_{occupiedi}.
Additionally, sampling error is ~N_{m}^{-1/2} when N_{m}
independent measurements are taken (i.e., N_{m} measurement cycles employing
all N probe units during each cycle), for N_{m} >> 1, establishing
an error bound on the probability of receptor occupancy of P_{occupiedi}
± N_{m}^{-1/2}. If two concentrations c_{1} = P_{occupiedi}
/ (1 - P_{occupiedi}) and c_{2} = (P_{occupiedi} + N_{m}^{-1/2})
/ (1 - (P_{occupiedi} + N_{m}^{-1/2})) may be distinguished,
where P_{occupiedi} is given by Eqn. 4.1,
then the minimum detectable concentration differential Dc
/c is

For k >> 1, Eqn. 4.2
reduces to: (Dc / c)_{large} ~ (1 + k^{1/2})
/ N_{m}^{1/2}. In the limit as k
approaches 1 and N_{m} >> 1, Eqn. 4.2
reduces to: (Dc / c)_{small} ~ k^{-1/2}
(Dc / c)_{large}.

For k = 10 (an N = 10-probe sensor,
sufficient to span the entire K_{d} = 10^{-4} to 10^{-13}
molecules/nm^{3} range using one probe per decade), Dc
/ c = 0.94 (94%) at N_{m} = 100, 0.20 (20%) at N_{m} = 1000,
(0.06) 6% at N_{m} = 10,000, and 0.006 (0.6%) at N_{m} = 10^{6}
measurement cycles. Little additional discrimination is obtained by using more
than N ~ 10 probes in the sensor. For N_{m} = 1000, Dc
/ c = 0.89 (89%) at k = 178 (N = 5 probes), 0.20
(20%) at k = 10 (N = 10 probes), 0.136 (13.6%) at
k = 1.23 (N = 100 probes), and 0.135 (13.5%) at k
= 1.0021 (N = 10,000 probes). Note that Dc / c =
0.20 (20%) is equivalent to detecting a pH variation of ~0.08 (e.g., 10^{(0.08)
}- 1 ~ 0.20).

Sensor cycle time ~t_{EQ} may be crudely approximated
by the number of random ligand-receptor encounters (N_{encounters} ~
100;^{ }Drexler^{10}) necessary
to ensure binding divided by the number of ligands striking the receptor surface
per second,^{434} or (sec/cycle)

where A = active receptor cross-sectional area ~0.1 nm^{2}
for small molecule (MW ~100 gm/mole) receptors to ~10 nm^{2} for large
molecule (MW ~10^{5} gm/mole) receptors; c_{ligand} = 10^{-2}
nm^{-3} for the most common molecules to 10^{-12} nm^{-3}
for rare molecules in the human body; MW_{kg} = target ligand molecular
weight in kg/mole, k = 0.01381 zJ/K (Boltzmann constant), T = 310 K (human body
temperature), and N_{A} = 6.023 x 10^{23} molecules/mole (Avogadro's
number).

For common small molecules, t_{EQ} ~10^{-6}
sec; for common large molecules, t_{EQ} ~0.2 x 10^{-6} sec,
but a sample chamber ~(25 nm)^{3} in size is diffusion-limited to a
minimum t_{EQ} ~ 10^{-5} sec for c_{ligand} **>~**
10^{-4 }nm^{-3}. For the rarest small molecules in the human
body, t_{EQ} ~ 800 sec, ~200 sec for rare large molecules, and a larger
chamber may also be required. Preconcentration of the test sample by dehydration
using sorting rotors (Section 3.4.2) to rapidly remove
water from the sample chamber reduces t_{EQ} to 10-40 sec for very rare
molecules.

Steric probes using a driving force^{10}
of ~100 pN, over 2 nm of motion per cycle require 200 zJ/measurement. If 1000
measurement cycles of a 10-probe sensor suffice to determine concentration to
~2 significant figures, this information costs ~0.002 pJ to acquire, a power
consumption of from ~10 pW for large common molecules to ~10^{-9} pW
(~0.6 kT/sec) for small rare molecules while the sensor is working. Power dissipation
may run up to ~10 times higher, depending upon the number of occupied receptors
that must be cleared via ejection rods during each cycle.

Last updated on 16 February 2003