Home > About Nanotechnology > Applications > Nanomedicine |

© Copyright 1998, Robert A. Freitas Jr.

All rights reserved.

Please send comments to

webmaster@foresight.org

Chapt. 3 Table of Contents | Page 1 | Page 2 | Page 3 | Page 4 |

The human body consists of ~7 x 10^{27} atoms arranged
in a highly aperiodic physical structure. Although 41 chemical
elements are commonly found in the body's construction (Table 3-1), CHON comprises 99% of its atoms.
Fully 87% of human body atoms are either hydrogen or oxygen.

**Table 3-1.**
Estimated Atomic Composition

of the Lean 70-kg Male Human Body

*(compiled & adapted from [749, 751-752, 817])*

Element | Sym | # of Atoms | Element | Sym | # of Atoms | Element | Sym | # of Atoms | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Hydrogen | H | 4.22 x 10^{27} |
Rubidium | Rb | 2.2 x 10^{21} |
Zirconium | Zr | 2 x 10^{19} |
||||||||

Oxygen | O | 1.61 x 10^{27} |
Strontium | Sr | 2.2 x 10^{21} |
Cobalt | Co | 2 x 10^{19} |
||||||||

Carbon | C | 8.03 x 10^{26} |
Bromine | Br | 2 x 10^{21} |
Cesium | Cs | 7 x 10^{18} |
||||||||

Nitrogen | N | 3.9 x 10^{25} |
Aluminum | Al | 1 x 10^{21} |
Mercury | Hg | 6 x 10^{18} |
||||||||

Calcium | Ca | 1.6 x 10^{25} |
Copper | Cu | 7 x 10^{20} |
Arsenic | As | 6 x 10^{18} |
||||||||

Phosphorus | P | 9.6 x 10^{24} |
Lead | Pb | 3 x 10^{20} |
Chromium | Cr | 6 x 10^{18} |
||||||||

Sulfur | S | 2.6 x 10^{24} |
Cadmium | Cd | 3 x 10^{20} |
Molybdenum | Mo | 3 x 10^{18} |
||||||||

Sodium | Na | 2.5 x 10^{24} |
Boron | B | 2 x 10^{20} |
Selenium | Se | 3 x 10^{18} |
||||||||

Potassium | K | 2.2 x 10^{24} |
Manganese | Mn | 1 x 10^{20} |
Beryllium | Be | 3 x 10^{18} |
||||||||

Chlorine | Cl | 1.6 x 10^{24} |
Nickel | Ni | 1 x 10^{20} |
Vanadium | V | 8 x 10^{17} |
||||||||

Magnesium | Mg | 4.7 x 10^{23} |
Lithium | Li | 1 x 10^{20} |
Uranium | U | 2 x 10^{17} |
||||||||

Silicon | Si | 3.9 x 10^{23} |
Barium | Ba | 8 x 10^{19} |
Radium | Ra | 8 x 10^{10} |
||||||||

Fluorine | F | 8.3 x 10^{22} |
Iodine | I | 5 x 10^{19} |
|||||||||||

Iron | Fe | 4.5 x 10^{22} |
Tin | Sn | 4 x 10^{19} |
|||||||||||

Zinc | Zn | 2.1 x 10^{22} |
Gold | Au | 2 x 10^{19} |
TOTAL |
6.71 x 10^{27} |

Somatic atoms are generally present in combined form as
molecules or ions, not individual atoms. The molecules of
greatest nanomedical interest are incorporated into cells or
circulate freely in blood plasma or the interstitial fluid. Table 3-2 summarizes the gross molecular
contents of the typical human cell, which is 99.5% water and
salts, by molecule count, and contains ~5000 different types of
molecules. Appendix B lists 261 of the most common molecular and
cellular constituents of human blood, and their normal
concentrations in whole blood and plasma. This listing is far
from complete. The human body is comprised of ~10^{5}
different molecular species, mostly proteinsa large but nonetheless finite molecular parts list.
By 1997, at least ~10^{4} of these proteins had been
sequenced, ~10^{3} had been spatially mapped, and ~7,000
structures (including proteins, peptides, viruses,
protein/nucleic acid complexes, nucleic acids, and carbohydrates)
had been registered in the Protein
Data Bank maintained at Brookhaven National Laboratory
[1144]. It is likely that the sequences and 3-D or tertiary
structures of all human proteins will have been determined by the
second decade of the 21st century, given the current accelerating
pace of improving technology [1145].

Transporting and sorting such a broad range of essential molecular species will be an important basic capability of many nanomedical systems. The three principal methods for distinguishing and conveying molecules that are most useful in nanomedicine are diffusion transport (Section 3.2), membrane filtration (Section 3.3), and receptor-based transport (Section 3.4). The chapter ends with a brief discussion of binding site engineering (Section 3.5).

**Table 3-2.**
Estimated Gross Molecular Contents

of a Typical 20-micron Human Cell

*(compiled and revised from [398, 531, 758-760, 938])*

Molecule | Mass % | MW (daltons) | # Molecules | Molecule % | Number of Molecular Types |
|||||
---|---|---|---|---|---|---|---|---|---|---|

Water | 65% | 18 | 1.74 x 10^{14} |
98.73 % | 1 | |||||

Other Inorganic | 1.5% | 55 | 1.31 x 10^{12} |
0.74 % | 20 | |||||

Lipid | 12% | 700 | 8.4 x 10^{11} |
0.475 % | 50 | |||||

Other Organic | 0.4% | 250 | 7.7 x 10^{10} |
0.044 % | ~200 | |||||

Protein | 20% | 50,000 | 1.9 x 10^{10} |
0.011 % | ~5,000 | |||||

RNA | 1.0% | 1 x 10^{6} |
5 x 10^{7} |
3 x 10^{-5}
% |
---- | |||||

DNA | 0.1% | 1 x 10^{11} |
46 | 3 x 10^{-11}
% |
---- | |||||

TOTALS |
100% |
---- |
1.76 x
10^{14} |
100% |
---- |
|||||

Chapt. 3 Table of Contents |

Fluidic transfer of material, known as convective-diffusive transport, can occur either by convection due to bulk flow or by diffusion due to Brownian motion. In convective transport, material is carried along fluid streamlines at the mean velocity of the fluid, with a velocity distribution such as that in Poiseuille flow (Section 9.4.1.X). Bulk flow is customarily regarded as the most important physiological transport mechanism in the human circulation. Only for the smallest molecules, such as water or glucose, does the time required to diffuse across the width of a capillary roughly equal the time taken by a fluid element to flow the same distance (~0.02 sec). Larger molecules such as fibrinogen take ~100 times longer (~2 sec) to diffuse across one capillary width.

However, bulk flow in the body is usually laminar. Transported materials travel parallel to (and thus cannot reach) fluid/solid interfaces such as the surfaces of blood vessels or membranes. Wall interactions are made possible by diffusion, a random process in which particles can move transversely to fluid streamlines in response to molecular-scale collisions.

Additionally, the movement of micron-scale devices within a bulk fluid flow is dominated by viscous, not inertial, forces (Section 9.4.1.X). Molecular transport to and from such nanodevices is governed by diffusion, not by bulk flow.

A particle suspended in a fluid is subjected to continuous collisions, from all directions, with the surrounding molecules. If the velocities of all molecules were the same all the time, the particle would experience no net movement. However, molecules do not have a single velocity at a given temperature, but rather have a distribution of velocities of varying degrees of probability. Thus from time to time, a suspended particle receives a finite momentum of unpredictable direction and magnitude. The velocity vector of the particle changes continuously, resulting in an observable random zigzag motion, called Brownian movement.

Einstein [385] approximated the RMS (root mean square) displacement of a particle of radius R suspended in a fluid of absolute viscosity and temperature T, after an observation period , as:

X
= (kT/ 3R)^{1/2} (meters) |
(3.1) |

where k = 1.381 x 10^{-23} joule/kelvin (K) or 0.01381
zJ/K (Boltzmann constant).^{1}
Particles under bombardment also experience a rotational Brownian
motion around randomly oriented axes, with the RMS angle of
rotation:

= (kT/ 4R^{3})^{1/2}
(radians) |
(3.2) |

although for < _{min} = M / 15R, where M is particle mass (see below), rotation
is ballistic.

In human blood plasma, with = 1.1 centipoise (1.1 x 10^{-3}
kg/m-sec) and T = 310 K, a spherical 1-micron diameter nanodevice
(R = 0.5 micron) translates ~1 micron in 1 sec (v_{brownian}
~ 10^{-6} m/sec) or ~8 microns (~the width of a
capillary) after 77 sec (v_{brownian} ~ 10^{-7}
m/sec), and rotates once in ~16 sec (_{min} = 2 x 10^{-8}
sec). In the same environment, a rigid 10-nm particle (roughly
the diameter of a globular protein) would translate ~8 microns in
one second (v_{brownian} ~ 10^{-5} m/sec) while
rotating ~250 times, due to Brownian motion (_{min}
= 2 x 10^{-12} sec).

The instantaneous thermal velocity over one mean free path
(the average distance between collisions) is much higher than the
net Brownian translational velocity would suggest. For a particle
of mass M = 4/3R^{3}
with mean (working) density , the mean thermal velocity is

v_{thermal} = (3kT/M)^{1/2} |
(3.3) |

At T = 310 K, a spherical 1-micron diameter nanodevice of
normal density (e.g. taking ~ _{H}_{2}_{O}
= 994.9 kg/m^{3} to minimize ballasting requirements;
Section 10.X.X) has v_{thermal} ~ 5 x 10^{-3}
m/sec; for a spherical 10-nm diameter protein with ~ 1500
kg/m^{3}, v_{thermal} ~ 4 m/sec.

^{1} The zeptojoule (zJ), or 10^{-21} joule, is the
standard unit of energy in the molecular nanotechnology
community; 1 zJ ~ 0.144 kcal/mole, the preferred unit among
chemists.

Chapt. 3 Table of Contents |

Medical nanodevices will frequently be called upon to absorb some particular material from the external aqueous operating environment. Molecular diffusion presents a fundamental limit to the speed at which this absorption can occur. (Once a block of solution has passed into the interior of a nanodevice, it may be divergently subdivided and transported at ~0.01-1 m/sec along internal pathways of characteristic dimension ~1 micron far faster than the <1 mm/sec diffusion velocity across 1 micron distances; Section 9.2.7.5.)

For a spherical nanodevice of radius R, the maximum diffusive intake current is

J = 4RDC | (3.4) |

where J is the number of molecules/sec presented to the entire
surface of the device, assumed to be 100% absorbed (but see
4.2.5), D (m^{2}/sec) is the translational Brownian
diffusion coefficient for the molecule to be absorbed, and C
(molecules/m^{3}) is the steady-state concentration of
the molecule far from the device [337]. (Blood concentrations in
gm/cm^{3} from Appendix B are converted to molecules/m^{3}
by multiplying Appendix B figures by (10^{6} x N_{A}/MW),
where N_{A} = 6.023 x 10^{23} molecules/mole
(Avogadro's number), MW = molecular weight in gm/mole or
daltons.) For rigid spherical particles of radius r, where r
>> r_{H}_{2}_{O}, the
Einstein-Stokes equation [387] gives

D = kT/(6r) | (3.5) |

though this is only an approximation because D varies slightly with concentration, with departure from molecular sphericalness, and other factors.

Measured diffusion coefficients in water for various molecules
of physiological interest, converted to 310 K, are in Table 3-3. (Diffusion coefficient data for
ionic salts such as NaCl and KCl, which dissociate in water and
diffuse as independent ions, are for solvated electrolytes.) A
1-micron (diameter) spherical nanodevice suspended in arterial
blood plasma at 310 K, with C = 7.3 x 10^{22} molecules/m^{3}
of oxygen and D = 2.0 x 10^{-9} m^{2}/sec,
encounters a flow rate of J = 9.2 x 10^{8} molecules/sec
of O_{2} impinging upon its surface. (The same
calculation applied to serum glucose yields J = 1.3 x 10^{10}
molecules/sec.) The characteristic time for change mediated by
diffusion in a region of size L scales as ~L^{2}/D (Eqn.
3.9, below). Across the diameter of an L = 1 micron nanodevice,
small molecules such as glucose diffuse in ~0.001 sec, small
proteins like hemoglobin in ~0.01 sec, and virus particles
diffuse in ~0.1 sec. (Diffusion coefficients of the same
molecules in air at room temperature are a factor of ~60 higher,
because _{air} ~ 183 micropoise at 20 °C.)

In blood, the diffusivity of larger particles is significantly
elevated because local fluid motions generated by individual red
cell rotation lead to greater random excursions of the particles
[388]. The effective diffusivity D_{e} = D + D_{r},
where the rotation-induced increase in diffusivity D_{r}
~ 0.25 R_{rbc}^{2}, with red cell radius R_{rbc}
~ 2.8 microns (taken for convenience as a spherical volume
equivalent) and a typical blood shear rate ~
500 sec^{-1}, giving D_{r} ~10^{-9} m^{2}/sec
in normal whole blood. The elevation of diffusivity caused by red
cell stirring is just 50% for O_{2} molecules. However,
for large proteins and viruses the effective diffusivity
increases 10-100 times, and the effective diffusivity of
particles the size of platelets is a factor of 10,000 higher than
for Brownian molecular diffusion.

The diffusion current to the surface of a nanodevice can also
be estimated for various nonspherical configurations [337]. For
instance, the diffusion current to both sides of an isolated thin
disk of radius R is given by J = 8RDC. The two-sided current to a
square thin plate of area L^{2} is J = (8/^{1/2}) LDC.
The steady-state diffusion current to an isolated cylinder of
length L_{c} and radius R is approximated by J = 2L_{c} DC/(ln (2L_{c}/R)
- 1), for L_{c} >> R. The diffusion current through
a circular hole of radius R in an impermeable wall separating
regions of concentration c_{1} and c_{2} is J =
4RD (c_{1}-c_{2}).

**Table 3-3.**
Translational Brownian Diffusion Coefficients for

Physiologically Important Molecules Suspended in Water at 310 K

*(most values converted from measured data at 20 °C, from
[390, 754, 761-763])*

Diffusing Particle in Water | Mol. Wt. (gm/mole) |
Diff. Coeff. D (m ^{2}/sec) |
||
---|---|---|---|---|

H_{2} |
2 | 5.4 x 10^{-9} |
||

H_{2}O |
18 | 2.31 x 10^{-9} |
||

O_{2} |
32 | 2.0 x 10^{-9} |
||

Methanol | 32 | 1.5 x 10^{-9} |
||

HCl | 36.5 | 3.6 x 10^{-9} |
||

CO_{2} |
44 | 1.9 x 10^{-9} |
||

NaCl | 58.5 | 1.5 x 10^{-9} |
||

Urea | 60 | 1.3 x 10^{-9} |
||

Glycine | 75 | 1.0 x 10^{-9} |
||

KCl | 75 | 2.0 x 10^{-9} |
||

-Alanine isomer | 89 | 9.5 x 10^{-10} |
||

-Alanine isomer | 89 | 9.7 x 10^{-10} |
||

Glycerol | 92 | 8.8 x 10^{-10} |
||

CaCl_{2} |
111 | 1.2 x 10^{-9} |
||

Glucose | 180 | 7.1 x 10^{-10} |
||

Mannitol | 182 | 7.1 x 10^{-10} |
||

Citric acid | 192 | 6.9 x 10^{-10} |
||

Sucrose | 342 | 5.4 x 10^{-10} |
||

Milk lipase | 6,669 | 1.5 x 10^{-10} |
||

Ribonuclease | 13,683 | 1.3 x 10^{-10} |
||

Insulin | 24,430 | 7.7 x 10^{-11} |
||

Scarlet fever toxin | 27,000 | 1.0 x 10^{-10} |
||

Somatotropin | 27,100 | 9.4 x 10^{-11} |
||

Carbonic anhydrase Y | 30,640 | 1.1 x 10^{-10} |
||

Plasma mucoprotein | 44,070 | 5.6 x 10^{-11} |
||

Ovalbumin | 43,500 | 8.1 x 10^{-11} |
||

Serum albumin | 68,460 | 6.5 x 10^{-11} |
||

Hemoglobin | 68,000 | 7.3 x 10^{-11} |
||

Transferrin | 74,000 | 6.2 x 10^{-11} |
||

Gonadotropin | 98,630 | 4.7 x 10^{-11} |
||

Collagenase | 109,000 | 4.5 x 10^{-11} |
||

Actin | 130,000 | 5.3 x 10^{-11} |
||

Plasminogen (profibrolysin) | 143,000 | 3.1 x 10^{-11} |
||

Ceruloplasmin | 143,300 | 5.0 x 10^{-11} |
||

-Globulin | 153,100 | 4.2 x 10^{-11} |
||

Immunoglobulin G (IgG) | 158,500 | 4.2 x 10^{-11} |
||

Hyaluronic acid | 177,100 | 1.3 x 10^{-11} |
||

Glucose dehydrogenase | 190,000 | 3.6 x 10^{-11} |
||

Fibrinogen | 339,700 | 2.1 x 10^{-11} |
||

Collagen | 345,000 | 7.3 x 10^{-12} |
||

Urease | 482,700 | 3.7 x 10^{-11} |
||

Cytochrome a | 529,800 | 3.8 x 10^{-11} |
||

-Macroglobulin | 820,000 | 2.5 x 10^{-11} |
||

-Lipoprotein | 2,663,000 | 1.8 x 10^{-11} |
||

Ribosome | 4,000,000 | 1.3 x 10^{-11} |
||

Viral DNA | 6,000,000 | 1.4 x 10^{-12} |
||

Urinary mucoprotein | 7,000,000 | 3.4 x 10^{-12} |
||

Tobacco mosaic virus | 31,340,000 | 5.6 x 10^{-12} |
||

T7 Bacteriophage | 37,500,000 | 9.5 x 10^{-12} |
||

Polyhedral silkworm virus | 916,200,000 | 2.3 x 10^{-12} |
||

1-micron spherical nanodevice | ~8 x 10^{11} |
4.1 x 10^{-13} |
||

Platelet (~2.4 microns) | ~4 x 10^{12} |
1.6 x 10^{-13} |
||

Red Blood Cell (~5.6 microns) | ~6 x 10^{13} |
6.8 x 10^{-14} |
||

Chapt. 3 Table of Contents |

Foraging nanodevices operating in aqueous environments may only modestly exceed the maximum rates of passive diffusive intake described in Section 3.2.2 by engaging in active physical movements designed to increase access to the desired molecules.

The first strategy for active diffusive intake is local
stirring. For this, the nanodevice is equipped with suitable
active appendages used to manipulate the fluid in its vicinity.
Transport by stirring is characterized by a velocity v_{a},
the speed of the appendage, and by a length L_{a}, its
distance of travel, which together define a characteristic
stirring frequency _{stir} ~ v_{a}/L_{a}
sec^{-1}. Movement of molecules over a distance L_{a}
by diffusion alone is scaled by a characteristic time ~L_{a}^{2}/D
(Section 3.2.2), which defines a
characteristic diffusion frequency _{diff}
~ D/L_{a}^{2} sec^{-1}. Stirring will be
more effective than diffusion only if _{stir}
> _{diff}, that is, if v_{a}
> D/L_{a}. For local stirring, L_{a} cannot be
much larger than the size of the nanodevice itself. Assuming L_{a}
= 1 micron and D = 10^{-9} m^{2}/sec for small
molecules, then v_{a} > 1000 microns/sec, a faster
motion than is exhibited by bacterial cells but quite modest for
nanomechanical devices (Section 9.3.1). With D = 10^{-11}
m^{2}/sec for large proteins and virus particles, v_{a}
> 10 microns/sec, well within the normal microbiological
range.

The ratio of stirring time to diffusion time, or Sherwood number

N_{Sh} = L_{a}v_{a}/D |
(3.6) |

provides a dimensionless measure of the effectiveness of
stirring vs. diffusion. For bacteria absorbing small molecules, N_{Sh}
~ 10^{-2}. Micron-scale nanodevices with 1-micron
appendages capable of 0.01-1 m/sec movement can achieve N_{Sh}
~ 10-1000 for small to large molecules, hence could be
considerably more effective stirrers.

In a classic paper, Berg and Purcell [337] analyzed the
viscous frictional energy cost of moving the stirring appendages
so that the fluid surrounding a spherical object (e.g. a
nanodevice) of radius R, out to some maximum stirring radius R_{s},
is maintained approximately uniform in concentration. The
objective is to transfer fluid from a distant region of
relatively high concentration to a place much closer to the
nanodevice, thereby increasing the concentration gradient near
the absorbing surface. To double the passive diffusion current by
stirring, the minimum required power density

P_{d} ~ (12D^{2}/R^{4})((R_{s}+2R)/(R_{s}-2R))^{3}
000 (watts/m^{3}) |
(3.7) |

If = 1.1 x 10^{-3} kg/m-sec, R = 0.5 micron, D
= 10^{-9} m^{2}/sec for small molecules, and
using a modest L_{a} = 1 micron stirring apparatus giving
R_{s} = 3R, then P_{d} ~ 3 x 10^{7}
watts/m^{3}. This greatly exceeds the 10^{2}-10^{6}
watts/m^{3} power density commonly available to
biological cells (Table 6-9) but lies well within the normal
range for nanomechanical systems which typically operate at up to
~10^{9} watts/m^{3}. (Nanomedically safe *in
vivo* power densities are discussed at length in Sections
6.5.2 and 6.5.3.) For D ~ 10^{-11} m^{2}/sec for
large molecules, P_{d} ~ 3 x 10^{3} watts/m^{3},
which is reasonable even by biological standards. The maximum
possible gain from stirring is ~ R_{s}/R, because the
current is ultimately limited to what can diffuse into the
stirred region.

Local heating due to stirring is minor. Given device volume V
~ 1 micron^{3}, P_{d} = 3 x 10^{7}
watts/m^{3}, mixing distance L_{mix} ~ 5 microns,
and thermal conductivity K_{t} = 0.623 watts/m-K for
water, then T ~ (P_{d}V/L_{mix}K_{t})
= 10 microkelvins; taking heat capacity C_{V} = 4.19 x 10^{6}
J/m^{3}-K for water, thermal equilibration time t_{EQ}
~ L_{mix}^{2}C_{V}/K_{t} = 0.2
millisec.

The second strategy for active diffusive intake is by
swimming. Again, the nanodevice is equipped with suitable active
propulsion equipment (Section 9.4) which enable it to move so as
to continuously encounter the highest possible concentration
gradient near its surface. Consider a spherical motile nanorobot
of radius R propelled at constant velocity v_{swim}
through a fluid containing a desired molecule for which the
surface of the device is essentially a perfect sink (Section
4.2.5). Applying the Stokes velocity field flow around the sphere
to the standard diffusion equation, a numerical solution by Berg
and Purcell [337] found that the fractional increase in the
diffusion current due to swimming is proportional to v_{swim}^{2}
for v_{swim} << D/R, and to v_{swim}^{1/3}
for v_{swim} >> D/R.

Diffusive intake is doubled at a swimming speed v_{swim}
= 2.5 D/R, which for 1-micron devices is ~5000 microns/sec when
absorbing small molecules, ~50 microns/sec for large molecules.
The viscous frictional energy cost to drive the nanodevice
through the fluid, derived from Stokes' law, requires an onboard
power density of

P_{d} = 9v_{swim}^{2}/2R^{2} |
(3.8) |

If = 1.1 x 10^{-3} kg/m-sec, v_{swim} =
2.5D/R, R = 0.5 micron, D = 10^{-9} m^{2}/sec for
small molecules, then P_{d} ~ 5 x 10^5 watts/m^{3}.
For large molecules with D = 10^{-11} m^{2}/sec,
P_{d} ~ 50 watts/m^{3}. Thus the energy cost of
diffusive swimming appears modest for nanomechanical systems;
gains in diffusion by swimming for nanodevices will be restricted
primarily by the maximum safe velocity that may be employed *in
vivo* (Section 9.4.X).

In general, outswimming diffusion requires movement over a
characteristic distance L_{s} ~ D/v_{swim} [389].
For bacteria moving at ~30 micron/sec and absorbing small
molecules, then L_{s} ~ 30 microns, roughly the sprint
distance exhibited by flagellar microbes such as E. coli. For
micron-scale nanodevices moving at ~1 cm/sec (Section 9.4.X), L_{s}
~ 1-100 nm for large to small molecules.

Chapt. 3 Table of Contents |

Nanodevices may also use diffusion to sort molecules. One of the remarkable features of diffusive sortation is that an input sample consisting of a complex mixture of many different molecular species can sometimes be completely resolved into pure fractions without having any direct knowledge of the precise shapes or electrochemical characteristics of the molecules being sorted. This can be a tremendous advantage for nanodevices operating in environments containing a large number of unknown substances. Another major advantage is the ability to readily distinguish isomeric (though not chiral) molecules. As one example of many possible, molecules suspended in water will diffuse into an adjacent region of pure water at different speeds, giving rise to dissimilar time-dependent concentration gradients which may be exploited for sortation by interrupting the process before complete diffusive equilibrium is reached.

For simplicity, assume we wish to separate two molecular
species initially present in solution in equal concentrations (c_{1}
= c_{2}), but having unequal diffusion coefficients (D_{1}
< D_{2}). Consider a separation apparatus with two
chambers. Chamber A contains input sample concentrate. Chamber B
contains pure water. A dilating gate (Section 3.3.2) separates
the two chambers. The gate is opened for a time t approximated by

t
= (X)^{2}/2D_{2}
~ L^{2}/2D_{2} |
(3.9) |

which relates the diffusion coefficient to the mean displacement X, taken here as L, the length of Chamber B. Table 3-4 gives an estimate of the time required for diffusion to reach 90% completion for glycine, a typical small molecule, in aqueous solution.

**Table 3-4.**
Estimated Time for Diffusion to Reach 90% Completion

for Glycine in Aqueous Solution at 310 K [397]

Diffusion Distance |
Diffusion Time (sec) |
Mean Velocity |
||
---|---|---|---|---|

1 nm | 10^{-9} |
1 m/sec | ||

10 nm | ~10^{-7} |
100 mm/sec | ||

100 nm | 10^{-5} |
10 mm/sec | ||

1 micron | 10^{-3} |
1 mm/sec | ||

10 microns | 10^{-1} |
100 micon/sec | ||

100 microns | 10 | 10 micron/sec | ||

1 mm | 1000 (17 min) | 1 micron/sec | ||

1 cm | 10^{5}
(28 hr) |
0.1 micron/sec |

After t has elapsed, the gate is
closed. (A gate with 10-nm sliding segments moving at 10 cm/sec
closes in 0.1 microsec.) The faster-diffusing component D_{2}
approaches diffusive equilibrium in Chamber B, but the
slower-diffusing component does not; it is present only in
smaller amounts. This gives a separation factor c_{2}/c_{1}
~ D_{2}/D_{1} for each diffusion sortation unit.
If n units are connected in series, with each unit receiving as
input the output of the previous unit, the net concentration
achieved by the entire cascade is ~(D_{2}/D_{1})^{n}.
Such cascades are commonplace in gaseous diffusion isotope
separation [875] and other applications.

Figure 3-1 shows a 2-dimensional
representation of an efficient design for a simple diffusion unit
that might be used in a sortation cascade. Each unit consists of
five chambers of equal volume, 7 dilating gates, 3 flap valves, 3
pistons, and two sieves which pass only water (or smaller)
molecules. Each chamber is roughly cubical with L ~ 35 nm along
the inside edge; including full piston throws and drives,
controls, interunit piping and other support structures, each
unit measures ~125 nm x 100 nm x 8 0 nm or ~0.001 micron^{3}
with a mass of ~10^{-18} kg.

Full size diagram of Diffusion
Cascade Sortation Unit, 682 x 482 pixels, 40K

The following is a precise description of one complete cycle of operation for each unit:

(1) The cycle begins with fluid to be sorted in Chamber A, Chambers B and W full of pure water with piston W all the way out, Chambers R and D empty with pistons R and D all the way in, and all valves and gates closed.

(2) Gate AB is opened for a time t, then closed. For a small molecule such as urea (MW = 60 daltons), t = 1 microsec; for a large molecule such as the enzyme urease (MW = 482,700 daltons), t = 35 microsec.

(3) Valves AI- and AI+, and gate AR, are opened. Piston R is drawn fully out, slowly to preserve laminar flow and to prevent mixing. Fluid in Chamber A is drawn into Chamber R. Fluid passing through the DO gate of the previous unit in the cascade enters Chamber A through valve AI-. Fluid passing through the RO valve of the subsequent unit in the cascade enters Chamber A through valve AI+. All valves and gates are closed; Chamber A is now ready for the next cycle.

(4) Gates WB and BD are opened. Piston W is slowly pushed all the way in while piston D is slowly pulled all the way out. Concentrated solution in Chamber B is transferred into Chamber D as pure water in Chamber W is transferred into Chamber B, again preserving laminar flow. Both gates are closed; Chamber B is now ready for the next cycle.

(5) Gates RW and DW are opened. Pistons R and D are slowly and simultaneously pushed halfway in while piston W is pulled all the way out. Forced at high pressure (~160 atm) through ~0.3 nm diameter sieve pores (Section 3.3.1), half of the solvent water present in Chambers R and D is pushed into Chamber W, filling Chamber W with water. (This design allows for easy backflushing if sieve pores become clogged.) Both gates are closed; Chamber W is now ready for the next cycle.

(6) Valve RO and gate DO are opened. Pistons R and D are slowly and simultaneously pushed the rest of the way in. Concentrated return fluid passes through valve RO and back to the AI+ input port of the previous unit in the cascade for further extraction. Concentrated diffusant fluid passes through gate DO and on to the AI- input port of the subsequent unit in the cascade for further purification. The valve and gate are closed; Chambers R and D are now empty and ready for the next cycle.

(7) Return to Step (1). (Adjacent units operate in counterphase while previous and subsequent units operate in synchrony, in a two-phase system.)

Increasingly purified sample passes through a multi-unit
sortation cascade as described above. For small molecules, a
cascade of n ~ 1000 units (total device volume ~1 micron^{3})
completely resolves two mixed molecular species with D_{2}/D_{1}
= 1.01. As a crude approximation, D ~ 1/MW^{1/3} for
small spherical particles [390], so this cascade separates small
molecules differing by the mass of one hydrogen atom, which
should be sufficient for most purposes. Structural isomeric forms
of the same molecule, such as -alanine
and -alanine, often have slightly different
diffusion coefficients, thus are also easily separable using a
diffusion cascade. However, stereoisomeric (chiral) forms cannot
be sorted by diffusion through an optically inactive solvent like
water.

For large molecules, a 1 million-unit cascade (total device
volume ~1000 micron^{3}) provides D_{2}/D_{1}
~ 1.00001, sufficient to completely separate large molecules
differing by the mass of a single carbon atom. The fidelity of
such fine resolutions depends strongly upon the ability to hold
constant the temperature of the chamber, since D varies directly
with temperature (Eqn. 3.5). Device
temperature stability will be determined by at least three
factors: (1) the accuracy of onboard thermal sensors in measuring
T (T/T < 10^{-6}; Section
4.6.1), (2) the rapidity with which the temperature measurement
can be taken (10^{-9} to 10^{-6} sec; Section
4.6.1), and (3) the time that elapses between the temperature
measurement and the end of the diffusive sortation process (which
may be of the same order as the gate closing time, ~10^{-6}
sec).

Most of the waste heat is generated in this device by forced
water sieving (Section 3.3.1). To remain within biocompatible
thermogenic limits (~10^{9} watts/m^{3}), each
unit may be cycled once every ~3 millisec, a 0.8% duty cycle of a
~23 microsec sieving stroke. Subject to this restriction, each
device would consume ~1 picowatt in continuous operation. A unit
presented with a ~0.1 M concentration of small molecules
processes ~10^{6} molecules/sec (e.g. ~1 gm/hour of
glucose using 1 cm^{3} of n = 1000-unit cascades), or ~10^{4}
molecules/sec for a unit presented with large molecules at ~0.001
M, circulating ~10^{9} molecules/sec of water as working
fluid while running at 340 cycles/sec. Additional chamber
segments on each unit, combined with more complex diffusion
circuits among the many units in a cascade, should permit the
simultaneous complete fractionation of the input feedstock even
if hundreds of distinct molecular species are present.

Chapt. 3 Table of Contents |

Nanoscale centrifuges offer yet another method for rapid molecular sortation, by biasing diffusive forces with a strong external field. The well-known effect of gravitational acceleration on spherical particles suspended in a fluid is described by Stokes' Law for sedimentation:

v_{t} = 2gR^{2}(_{particle}
- _{fluid})/9 |
(3.10) |

where v_{t} is terminal velocity, g is the
acceleration of gravity (9.81 m/sec^{2}), R is particle
radius, _{particle} and _{fluid} are the particle
and fluid densities (kg/m^{3}), and is
coefficient of viscosity of the fluid. Particles which are more
dense than the suspending liquid tend to fall. Those which are
less dense tend to rise (_{particle}/_{fluid}
~ 0.8 for lipids, up to ~1.5 for proteins, and ~1.6 for
carbohydrates).

This separation process may be greatly enhanced by rapidly spinning the mixed-molecule sample in a nanocentrifuge device. For ideal solutions (e.g. obeying Raoult's law) at equilibrium [390]:

(3.11) |

where c_{2} is the concentration at distance r_{2}
from the axis of a spinning centrifuge (molecules/m^{3}),
c_{1} is the concentration at distance r_{1}
(nearer the axis), MW_{kg} is the molecular weight of the
desired molecule in kg/mole, is the angular velocity of
the vessel (rad/sec), T is temperature (K) and the universal gas
constant R_{g} = 8.31 joule/mole-K. The approximate
spinning time t_{s} required to reach equilibrium is

t_{s} = ln(r_{2}/r_{1})/^{2} S_{d} |
(3.12) |

where S_{d} is the sedimentation coefficient, usually
given in units of 10^{-13} sec or svedbergs (Table 3-5). Research ultracentrifuges have
reached accelerations of ~10^{9} g's.

**Table 3-5.**
Sedimentation Coefficients for Particles

in Aqueous Suspension at 310 K

*(1 svedberg = 10*^{-13}* sec; values
converted from
measured data at 20 °C, from [390, 754])*

Particle | Diffusion Sed. Coeff. (sec) |
Mol. Wt. | ||
---|---|---|---|---|

O_{2} |
0.12 x 10^{-13} |
32 | ||

CO_{2} |
0.07 x 10^{-13} |
44 | ||

Glucose | 0.18 x 10^{-13} |
180 | ||

Insulin monomer | 1.5 x 10^{-13} |
6,000 | ||

Ribonuclease | 1.75 x 10^{-13} |
13,683 | ||

Lysozyme | 2.03 x 10^{-13} |
17,200 | ||

Insulin | 1.84 x 10^{-13} |
24,430 | ||

Ovalbumin | 3.4 x 10^{-13} |
43,500 | ||

Serum albumin | 4.3 x 10^{-13} |
68,460 | ||

Alcohol dehydrogenase | 7.2 x 10^{-13} |
150,000 | ||

Catalase | 10.7 x 10^{-13} |
250,000 | ||

-Lipoprotein | 5.6 x 10^{-13} |
2,663,000 | ||

Actomycin | 11.3 x 10^{-13} |
3,900,000 | ||

TMV | 175 x 10^{-13} |
31,340,000 |

Consider a cylindrical diamondoid vessel of density _{vessel}
= 3510 kg/m^{3}, radius r_{c} = 200 nm, height h
= 100 nm, and wall thickness x_{wall} = 10 nm, securely
attached to an axial drive shaft of radius r_{a} = 50 nm
(schematic in Figure 3-2). A fluid sample
containing desired molecules enters the vessel through a hollow
conduit in the drive shaft, and the device is rapidly spun. If
rim speed v_{r} = 1000 m/sec (max), then = v_{r}/r_{c}
= 5 x 10^{9} rad/sec (/2 = 8 x 10^{8} rev/sec).
The maximum bursting force F_{b} ~ 0.5 _{vessel}
v_{r}^{2} = 2 x 10^{9} N/m^{2},
well below the 50 x 10^{9} N/m^{2} diamondoid
tensile strength conservatively assumed by Drexler [10]. Since S_{d}
ranges from 0.1-200 x 10^{-13} sec for most particles of
nanomedical interest (Table 3-5), minimum
separation time using acceleration a_{r} / g = v_{r}^{2}/g
r_{c} = 5 x 10^{11} g's, when r_{2} = r_{c}
and r_{1} = r_{a}, is t_{s} = 0.003-6.0 x
10^{-6} sec. Fluid sample components migrate at ~0.1
m/sec.

**Figure 3-2. Teragravity Nanocentrifuge**

Maximum centrifugation energy per particle E_{c} = (MW_{kg}
/ N_{A}) a_{r} (r_{c}-r_{a}) ~
10,000 zJ/molecule, or ~10 zJ/bond for proteins, well below the
180-1800 zJ/bond range for covalent chemical bonds (Section
3.5.1). However, operating the nanocentrifuge at peak speed may
disrupt the weakest noncovalent bonds (including hydrophobic,
hydrogen, and van der Waals) which range from 4-50 zJ/bond. The
nanocentrifuge has mass ~10^{-17} kg, requires ~3
picojoules to spin up to speed (bearing drag consumes ~10
nanowatts of power, and fluid drag through the internal plumbing
contributes another ~ 5 nanowatts), completes each separation
cycle in ~10^{4} revs (~10^{-5} sec), and
processes ~300 micron^{3}/sec which is ~10^{13}
small molecules/sec (at 1% input concentration) or ~10^{9}
large molecules/sec (at 0.1% input concentration).

From Eqn. 3.11, the nanocentrifuge
separates salt from seawater with c_{2}/c_{1} ~
300 across the width of the vessel (r_{c} - r_{a}
= 150 nm); extracting glucose from water at 310 K, c_{2}/c_{1}
~10^{5} over 150 nm. For proteins with _{particle}
~ 1500 kg/m^{3}, separation product removal ports may be
spaced, say, 10 nm apart along the vessel radius while
maintaining c_{2}/c_{1} ~ 10^{3} between
each port. Vacuum isolation of the unit in an isothermal
environment and operation in continuous-flow mode could permit
exchange of contents while the vessel is still moving, sharply
reducing remixing, vibrations, and thermal convection currents
between product layers. A complete design specification of
product removal ports, batch and continuous flow protocols,
compression profiles, etc. is beyond the scope of this book.

Variable gradient density centrifugation may be used to trap
molecules of a specific density in a specific zone for subsequent
harvesting, allowing recovery of each molecular species from
complex mixtures of substances that are close in density. The
traditional method is a series of stratified layers of sucrose or
cesium chloride solutions that increase in density from the top
to the bottom of the tube. A continuous density gradient may also
be used, with the density of the suspension fluid calibrated by
physical compression. For example, the coefficient of isothermal
compressibility = - (V_{l} / V_{l} ) / P_{l} = ( _{fluid} / _{fluid})
/ P_{l} = 4.492 x 10^{-5}
atm^{-1} for water at 1 atm and 310 K (compressibility is
pressure- and temperature-dependent). Applying P_{l} =
12,000 atm to the vessel raises fluid density to 1250 kg/m^{3}
[567], sufficient to partially regulate protein zoning. A
multistage cascade (Section 3.2.4) may be
necessary for complete compositional separation. Protein
denaturation between 5000-15,000 atm [585] due to hydrogen bond
disruption may limit nanocentrifugation rotational velocity.
Protein compressibility may further reduce separability. The
balance between the differential densities and the differential
compressibilities will determine the equilibrium radius of the
protein in the centrifuge; in the limiting case of equal
compressibilities for a given target protein and water, there is
no stable equilibrium radius.

The nanocentrifuge may also be useful in isotopic separations.
For a D_{2}O/H_{2}O mixture, c_{2}/c_{1}
= 1.415 per pass through the device; c_{2}/c_{1}
= 10^{6} is achieved in a 40-unit cascade. Tracer glycine
containing one atom of ^{14}C is separated from natural
glycine using a 113-unit cascade, achieving c_{2}/c_{1}
= 10^{6}.

Chapt. 3 Table of Contents | Page 1 | Page 2 | Page 3 | Page 4 |

© Copyright 1998, Robert A. Freitas Jr. All rights reserved. |