Nanomedicine

by Robert A. Freitas Jr.

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

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Chapter 3. Molecular Transport and Sortation

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

3.1 Human Body Chemical Composition

The human body consists of ~7 x 10²⁷ 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])

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 ~10⁵ different molecular species, mostly proteinsa large but nonetheless finite molecular parts list. By 1997, at least ~10⁴ of these proteins had been sequenced, ~10³ 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])

Chapt. 3 Table of Contents

3.2 Diffusion Transport

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.

3.2.1 Brownian Motion

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/ 4R3)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 = 4/3R³ with mean (working) density , the mean thermal velocity is

At T = 310 K, a spherical 1-micron diameter nanodevice of normal density (e.g. taking ~ _H2O = 994.9 kg/m³ 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/m³, 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

3.2.2 Passive Diffusive Intake

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 (m²/sec) is the translational Brownian diffusion coefficient for the molecule to be absorbed, and C (molecules/m³) is the steady-state concentration of the molecule far from the device [337]. (Blood concentrations in gm/cm³ from Appendix B are converted to molecules/m³ by multiplying Appendix B figures by (10⁶ x N_A/MW), where N_A = 6.023 x 10²³ molecules/mole (Avogadro's number), MW = molecular weight in gm/mole or daltons.) For rigid spherical particles of radius r, where r >> r_H2O, 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 10²² molecules/m³ of oxygen and D = 2.0 x 10^-9 m²/sec, encounters a flow rate of J = 9.2 x 10⁸ molecules/sec of O₂ impinging upon its surface. (The same calculation applied to serum glucose yields J = 1.3 x 10¹⁰ molecules/sec.) The characteristic time for change mediated by diffusion in a region of size L scales as ~L²/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_rbc², 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 m²/sec in normal whole blood. The elevation of diffusivity caused by red cell stirring is just 50% for O₂ 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 L² 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 c₁ and c₂ is J = 4RD (c₁-c₂).

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 (m2/sec)
H2		2		5.4 x 10-9
H2O		18		2.31 x 10-9
O2		32		2.0 x 10-9
Methanol		32		1.5 x 10-9
HCl		36.5		3.6 x 10-9
CO2		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
CaCl2		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 1011		4.1 x 10-13
Platelet (~2.4 microns)		~4 x 1012		1.6 x 10-13
Red Blood Cell (~5.6 microns)		~6 x 1013		6.8 x 10-14

Chapt. 3 Table of Contents

3.2.3 Active Diffusive Intake

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.

3.2.3.1 Diffusive Stirring

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_a²/D (Section 3.2.2), which defines a characteristic diffusion frequency _diff ~ D/L_a² 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 m²/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 m²/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

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

Pd ~ (12D2/R4)((Rs+2R)/(Rs-2R))3  000 (watts/m3)

(3.7)

If = 1.1 x 10^-3 kg/m-sec, R = 0.5 micron, D = 10^-9 m²/sec for small molecules, and using a modest L_a = 1 micron stirring apparatus giving R_s = 3R, then P_d ~ 3 x 10⁷ watts/m³. This greatly exceeds the 10²-10⁶ watts/m³ 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 ~10⁹ watts/m³. (Nanomedically safe in vivo power densities are discussed at length in Sections 6.5.2 and 6.5.3.) For D ~ 10^-11 m²/sec for large molecules, P_d ~ 3 x 10³ watts/m³, 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 micron³, P_d = 3 x 10⁷ watts/m³, mixing distance L_mix ~ 5 microns, and thermal conductivity K_t = 0.623 watts/m-K for water, then T ~ (P_dV/L_mixK_t) = 10 microkelvins; taking heat capacity C_V = 4.19 x 10⁶ J/m³-K for water, thermal equilibration time t_EQ ~ L_mix²C_V/K_t = 0.2 millisec.

3.2.3.2 Diffusive Swimming

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_swim² 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

Pd = 9vswim2/2R2

(3.8)

If = 1.1 x 10^-3 kg/m-sec, v_swim = 2.5D/R, R = 0.5 micron, D = 10^-9 m²/sec for small molecules, then P_d ~ 5 x 10^5 watts/m³. For large molecules with D = 10^-11 m²/sec, P_d ~ 50 watts/m³. 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

3.2.4 Diffusion Cascade Sortation

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 (c₁ = c₂), but having unequal diffusion coefficients (D₁ < D₂). 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)2/2D2 ~ L2/2D2

(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]

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 D₂ 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 c₂/c₁ ~ D₂/D₁ 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 ~(D₂/D₁)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 micron³ 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 micron³) completely resolves two mixed molecular species with D₂/D₁ = 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 micron³) provides D₂/D₁ ~ 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 (~10⁹ watts/m³), 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 ~10⁶ molecules/sec (e.g. ~1 gm/hour of glucose using 1 cm³ of n = 1000-unit cascades), or ~10⁴ molecules/sec for a unit presented with large molecules at ~0.001 M, circulating ~10⁹ 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

3.2.5 Nanocentrifugal Sortation

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:

vt = 2gR2(particle - fluid)/9

(3.10)

where v_t is terminal velocity, g is the acceleration of gravity (9.81 m/sec²), R is particle radius, _particle and _fluid are the particle and fluid densities (kg/m³), 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 c₂ is the concentration at distance r₂ from the axis of a spinning centrifuge (molecules/m³), c₁ is the concentration at distance r₁ (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

ts = ln(r2/r1)/2 Sd

(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 ~10⁹ 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.
O2		0.12 x 10-13		32
CO2		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/m³, 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 10⁹ rad/sec (/2 = 8 x 10⁸ rev/sec). The maximum bursting force F_b ~ 0.5 _vessel v_r² = 2 x 10⁹ N/m², well below the 50 x 10⁹ N/m² 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_r²/g r_c = 5 x 10¹¹ g's, when r₂ = r_c and r₁ = 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 ~10⁴ revs (~10^-5 sec), and processes ~300 micron³/sec which is ~10¹³ small molecules/sec (at 1% input concentration) or ~10⁹ large molecules/sec (at 0.1% input concentration).

From Eqn. 3.11, the nanocentrifuge separates salt from seawater with c₂/c₁ ~ 300 across the width of the vessel (r_c - r_a = 150 nm); extracting glucose from water at 310 K, c₂/c₁ ~10⁵ over 150 nm. For proteins with _particle ~ 1500 kg/m³, separation product removal ports may be spaced, say, 10 nm apart along the vessel radius while maintaining c₂/c₁ ~ 10³ 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/m³ [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 D₂O/H₂O mixture, c₂/c₁ = 1.415 per pass through the device; c₂/c₁ = 10⁶ is achieved in a 40-unit cascade. Tracer glycine containing one atom of ¹⁴C is separated from natural glycine using a 113-unit cascade, achieving c₂/c₁ = 10⁶.

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