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by Robert A. Freitas Jr.

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

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Chapter 3. Molecular Transport and Sortation


3.3 Membrane Filtration

Filtration through a permeable membrane is closely related to the process of diffusion, since in both cases random molecular motions help carry the process to completion. However, the presence of a membrane adds a new measure of control that is not exploited in simple diffusive transport. This control may be either passive or active, as described below.


3.3.1 Simple Nanosieving

Nanometer-scale isoporous molecular sieves (with ovoid, square, or hexagonal holes) are common in almost every taxonomic group of eubacteria and archaeobacteria [525]. Other well-known examples of nanoporous structures are the 6-nm pore arrays found in reverse osmosis and (kidney) dialysis membranes.

Likewise, it is possible for a nanodevice to sort molecules by simple sieving [987, 1177]. In this process, a sample containing particles of various sizes suspended in water passes through a graduated series of filters perforated by progressively smaller holes of fixed size and shape. Between each filtration unit, the filtration residue consists almost exclusively of particles having a narrow range of sizes and shapes. For example, a series of n=100 filtration units could reliably differentiate an input sample containing particles from 0.2-1.2 nm into 100 separate fractions, each fraction consisting predominantly of particles differing in mean diameter by ~0.01 nm. (In a practical system, several passes would be required to achieve complete discrimination; Section 3.2.4.) A ~0.01 nm difference in molecular diameter corresponds to the mean contribution of ~1 additional carbon atom to the size of a small molecule (MW ~ 100 daltons), or to the mean contribution of ~100 additional carbon atoms to the size of a large molecule (MW ~ 100,000 daltons, ~17,000 atoms). A nanomembrane might even permit the (slow, multi-pass) sieving of oxygen from air, since the molecular diameters of N2 and O2 differ by ~0.01 nm (Section 3.5.5).

Two opposing forces are at work when moving water and solutes through a membrane. One is the osmotic pressure established by the presence of nonpermeating solutes; the other is the hydraulic or fluid pressure. The velocity of material movement depends on the relative values of the osmotic and hydraulic forces, and on the size of the pores in the filter.

Osmotic pressure ppi is given by the Donnan-van't Hoff formula

ppi = (Rg T (c2 - c1) / MWkg) (1 + Z2 (c2 - c1) / cs) (N/m2)   (3.13)

where Rg = 8.31 joules/mole-K, T is temperature in kelvins (K), c2 and c1 are solute concentrations on either side of the membrane in kg/m3 (c2 > c1), MWkg is the molecular weight of the solute in kg/mole, and the Z2 term (dependent upon polymer-polymer interactions) is a correction factor for highly concentrated solutions which for some solvents and temperatures may equal zero; Z = net solute charge number and cs is the concentration in kg/m3 of a second solute, as for example when the first solute is a protein and the second solute is salt, as in human serum [403]. Water at 310 K dissolves a maximum of c2 = 370 kg/m3 of sodium chloride (a 37.0% solution, by weight), and MWkg = 0.05844 kg/mole for NaCl, so for salt water solutions the theoretical maximum ppi ~ 1.6 x 107 N/m2 ~ 160 atm. Natural bloodstream concentrations of salt produce ppi ~ 3 atm. Since osmotic pressure depends on the number of molecules present, the contribution from large molecules is usually negligible. For instance, total protein concentration in human blood serum is c2 ~ 73 kg/m3, MWkg ~ 50 kg/mole (~50,000 daltons), so ppi ~ 0.04 atm.

In theory, extremely large hydraulic counterforces up to 105 atm may be applied in nanomechanical systems, for example by a piston, to overcome osmotic backpressure. As a practical matter, however, rapidly pushing small molecules at high pressure through nanoscale holes is an effective method for generating significant amounts of waste heat. A design compromise is required.

Consider a simple sorting apparatus in which a square piston is used to compress solvent fluid (say, water) trapped in a chamber hchamber in length and Lchamber2 in cross-sectional area, forcing the fluid to filter through pores of radius rpore (~ target molecule radius) covering a fraction (alpha)H (~50%) of the surface of a square nanoscale sieve of thickness hsieve and area Lsieve2. Solute which is dissolved or suspended in the solvent is sorted based on molecular size; a sequence of sieving runs using sieves having progressively smaller pore radii produces an ordered sequence of molecular size fractions. (In small molecules, adding the mass of one hydrogen atom increases the mean linear dimension of the molecule by 0.1-1%.)

The first design constraint on this system relates to its maximum operating pressure. If (delta)P is applied pressure (N/m2), then avoiding boiling the solvent water and denaturing proteins requires at least that (delta)Pmax < CV(delta)Tboil, where the heat capacity of water CV = 4.19 x 106 joules/m3-K and (delta)Tboil = 373 K - 310 K = 63 K give (delta)Pmax < 2600 atm. The designs presented below operate at 6% of this maximum ((delta)T ~ 3 K) or less.

There are two major design constraints on the duration of the power stroke, or tp:

(I) Molecular Rotation Constraint. Flow through the sieve must be slow enough to allow molecules to align with the holes. Assuming round pores, the total number of pores in the sieve is Npore = (alpha)H Lsieve2 / pi rpore2. The volume processing rate (m3/sec) of the sieve V dotsieve = V dotchamber = hchamber Lchamber2 / tp, hence the molecule processing rate is N dot = V dotchamber ctarget (molecules/sec) where ctarget is the concentration of target molecules (molecules/m3). At any one time, each pore channel through the sieve can hold at most Nchannel = hsieve / 2 rpore molecules in single file; during each power stroke, at most Nstack = N dot tp / Npore molecules pass through each pore. Hence the time available for molecular rotation trot = tp (Nchannel / Nstack), which assumes the layer of rotating target molecules in the vicinity of the pores approximates sieve thickness hsieve, a reasonable assumption as long as the typical molecular diffusion time (across a distance hsieve) << trot. Taking Nrot ~ 10 as the mean number of molecular revolutions needed to ensure proper pore alignment with noncircular sieve holes (the most difficult case), then from Eqn. (3.2) (delta)(alpha) = (kT trot / 4 pieta rpore3)1/2 > ~2 pi Nrot, where eta is solvent viscosity (1.1 x 10-3 kg/m-sec for plasma) at T = 310 K. Solving for minimum tp gives:

tp greater than or equal 32 pi4 Nrot2 eta ctarget hchamber rpore6 / kT (alpha)H hsieve (sec)   (3.14)

(II) Pressure/Flow Constraint. Flow through the sieve must be fast enough to establish a sufficient pressure to oppose osmotic backflows. From the Hagen-Poiseuille law (Section 9.2.5), the volume processing rate through each pore is V dotpore = pi rpore4 (delta)Psieve / 8 eta hsieve and the volume processing rate through the entire sieve is V dotsieve = Npore V dotpore = V dotchamber; solving for maximum tp gives:

tp less than or equal 8 eta hchamber hsieve Lchamber2 / (alpha)H (delta)Psieve rpore2 Lsieve2 (sec)   (3.15)

Equating these two bracketing constraints, hsieve greater than or equal 150 nm for large molecules (rpore ~ 5 nm) but hsieve greater than or equal 1 nm for small molecules (rpore ~ 0.32 nm).

As a final constraint, power released by fluid flow through the chamber and sieve must not exceed safe thermogenic limits. Given the maximum safe power density for in vivo nanomachines given in Section 6.5.3 as Dn = 109 watts/m3, then

Ddevice = Pdevice / (hchamber + hsieve) Lchamber2 less than or equal Dn (watts/m3)   (3.16)

where Ddevice = device power density (watts/m3), total device power Pdevice = Pchamber + Psieve (watts), chamber fluid flow power Pchamber = pi Lsieve4 (delta)Pchamber2 / 128 hchamber eta, sieve fluid flow power Psieve = (alpha)H Lsieve2 rpore 2 (delta)Psieve2 / 8 hsieve eta, and (delta)Pchamber = 16 (alpha)H hchamber rpore2 (delta)Psieve / pi Lsieve2 hsieve. For small molecules (e.g. NaCl or glucose), (delta)Psieve greater than or equal 160 atm to overcome maximum osmotic backpressure; for large molecules (e.g. ~50,000 dalton proteins), we assume (delta)Psieve greater than or equal 1 atm to ensure sieving. The following designs are not optimized but illustrate the tradeoffs involved.

For small molecules (rpore ~ 0.32 nm), an exemplar ~1 micron3 device has hchamber = 1 micron, Lchamber = Lsieve = 0.6 micron, hsieve = 1.5 microns, tp = 0.016 sec, (delta)Psieve = 160 atm, (delta)Pchamber = 0.0001 atm. Piston velocity ~ 60 micron/sec and V dot = 2.5 x 10-17 m3/sec for a 0.1 M solution of target molecules, yielding a processing rate of 1.5 x 109 molecules/sec (1.5 x 10-16 kg/sec); the device processes its own mass every ~7 sec or every ~430 power strokes. Device power Pdevice = 400 pW and power density Ddevice = 4 x 108 watts/m3.

For large molecules (rpore ~ 5.0 nm), an exemplar ~1 micron3 device has hchamber = 1 micron, Lchamber = Lsieve = 0.9 micron, hsieve = 0.15 microns, tp = 0.010 sec, (delta)Psieve = 1 atm, (delta)Pchamber = 0.0004 atm. Piston velocity ~ 100 micron/sec and V dot = 9.8 x 10-17 m3/sec for a 0.001 M solution of target molecules, yielding a processing rate of 5.9 x 107 molecules/sec (4.9 x 10-15 kg/sec); the device processes its own mass every ~0.2 sec or every ~20 power strokes. Device power Pdevice = 80 pW and power density Ddevice = 8 x 107 watts/m3.

Sieve pores can become clogged by particles of radius R ~ rpore if the applied hydraulic pressure (delta)Pclog exceeds the thermal energy of the trapped particles, that is, if

(delta)Pclog > 9 kT / 8 pi R3 (N/m2)   (3.17)

By this criterion, (delta)Pclog > 500 atm for small molecules and (delta)Pclog > 0.1 atm for large molecules. From the values of (delta)Psieve given above, clogging is unlikely for small molecules but is possible at the highest concentrations of large molecules. In 310 K water, large molecules diffuse ~3 nm and small molecules diffuse ~17 nm in ~10-7 sec, just far enough to clear the hole, so a ~10 MHz sawtooth pressure profile imposed on the power stroke should ensure sufficient backflushing action to avoid serious blockages. To reduce the possibility of clogging due to surface force adhesion (Section 9.2.3), as a design criterion the work of adhesion should be reduced to Wadhesion < (delta)Psieve rpore ~ 5 x 10-3 J/m2 for small molecules and ~ 0.5 x 10-3 J/m2 for large molecules likely to come into contact with sieve pore surfaces. Clogging due to long-term random polymerizations can be minimized by periodically exchanging the entire contents of the input chamber with fresh solution, by operating the device at reduced power density, or by periodically replacing the sieve.


3.3.2 Dynamic Pore Sizing

A more efficient nanosieve system can be designed if pore size and shape can be actively modified during device operation, as for example by exchanging filters (from a membrane library stocking various pore sizes) each half cycle. Better, if pores can be reliably dilated or constricted in place during a period of time (delta)t << tp, then filtration cascades can be more rapidly reconfigured to match changing input feedstock characteristics or to extract varying selections of desired molecules at will. Additionally, fully differentiating sieving cascades can be collapsed into a single unit, providing more compact devices especially useful in chemical sensor systems requiring preconcentration of sample (Section 4.2.1). Control of pore shape should also provide finer discrimination among molecules of similar size but different shape, such as some isomers of nonchiral molecules.

Two or more overlapping surfaces containing regular arrays of perforations of fixed size and shape can conveniently generate a wide variety of pore geometries. Control of pore geometry is achieved by sliding or rotating one surface relative to the other surface by a small increment, as suggested schematically by the examples in Figure 3-3. Circular dilating apertures can also be constructed using a matched set of overlapping segments, which may be driven either radially or tangentially to enlarge or contract the hole like an irising camera diaphragm (Figure 3-4). Diaphragming mechanisms may be vertically staggered to maximize areal hole density in filtration surfaces (at the cost of increased vertical rugosity). Filters constructed of hydrogen-passivated diamondoid can have pores with <0.1 nm feature sizes, although H-free fullerene materials would avoid any possibility of dehydrogenation shearing. Methods of positioning surfaces to accuracies of ~0.01 atomic diameter (~0.001 nm) are discussed in Section 3.5.6. Assuming pore sizing blades require ~25 nm2 of diamondoid contact surface per pore and each blade travels 25 nm at 0.01 meter/sec during one cycle, sliding friction [10] dissipates ~0.01 zJ/pore, or ~4 x 10-18 watts/pore during each 2.5 microsec resizing cycle. Since fluid friction approaches kT for nanometer-size holes changing size in ~10-9 sec, maximum blade speed is ~1 m/sec and the fastest resizing cycle is ~10-8 sec.

Variable Size/Shape Apertures

Figure 3-3. Variable Size/Shape Apertures Using Two Nanoscale Perforated Sliding Plates


Circular Dilating Diaphragm

Figure 3-4. Circular Dilating "Iris" Diaphragm Mechanism for Dynamic Nanopore Sizing

A single sieving unit with controllable pores can be moderately efficient. Consider a design similar to that described in Section 3.3.1, except for a separate chamber and piston on either side of the filter block. Suppose that the particles desired to be extracted are of radius r, and the next smallest possible pore size is r - (delta)r. The device operates in two phases. In the first phase, the sample is placed in the first chamber, pore size is set equal to r, then the first piston forces the fluid through the membrane. Particles larger than r remain behind and are flushed from the first chamber. The pores then contract to r - (delta)r, and the second piston pushes the remaining filtrate back into the first chamber. After this second phase, particles of radius ~(r ± (delta)r) remain in the second chamber at significantly higher concentration and may be removed for further use.

Other designs might work equally well, such as a 3-chamber flowthrough design using a variable pore membrane with pores of size r between the first and second chamber, a membrane with pores of size r - (delta)r between the second and third chamber, and a piston at either end (one pushing, one pulling), thus concentrating molecules of size r ± (delta)r in the central chamber. M. Krummenacker suggests fixed chambers with a moving sieve operated as a dragnet. Filtration processes may be most useful in performing complete separations of complex mixtures. But they are inefficient in the sense that the energy expended to orient molecules passing through pores is wasted if the molecules are allowed to randomize on the other side; a eutactic mill-like molecule handling system (Section 3.4.3) might preserve this order and greatly improve energy efficiency.


3.3.3 Gated Channels

Besides controlling nanopore size and shape, individual molecular transport channels can be gated either mechanically (e.g. ligand gating) or electrically (e.g. voltage gating) [1050]. Either method might usefully be employed to control molecular transport through the surfaces of medical nanodevices in a process that could very loosely be described as molecular transistor gating.

A good example of mechanical gating in biology is the nicotinic acetylcholine receptor channel, probably the best understood ligand-gated channel [391, 396]. Nerve impulses are communicated across neuromuscular junctions and autonomic ganglia via neurotransmitters such as acetylcholine. STM images [419] confirm that the receptor itself is cylindrical, a bundle of 5 rod-shaped polypeptide subunits arranged like barrel staves with outside diameter ~6.5 nm. The receptor protrudes 6 nm on the synaptic side of the membrane and 2 nm on the cytoplasmic side. The water-filled channel pore lies along the symmetry axis, lined by 5 (alpha)-helices, with a 2.2 nm wide mouth on the synaptic surface, a 0.65 nm waist where the structure dives through the cell membrane, and a 2 nm wide cytosolic exit.

Normally, the channel is closed and no ions may pass. In this closed state, the channel is occluded at the waist by a ridge of large residues forming a tight hydrophobic ring. Each subunit has a bulky leucine at the bend in the (alpha)-helix, a critical position. When two acetylcholine molecules bind to the receptor, these helices allosterically tilt, shifting the position of the ridges. The pore becomes open because it is now lined with small polar residues rather than by large hydrophobic ones. This conformational change allows 2.5 x 107 Na+ ions/sec to flow through the channel, about 10% of the diffusion-limited rate. (Anions like Cl- cannot enter the pore because they are repelled by rings of negatively charged residues positioned at either end of the receptor.)

Acetylcholine binding opens the gate in less than 100 microsec under physiologic conditions. Subsequent rapid destruction of acetylcholine by acetylcholinesterase, an enzyme tethered to the membrane surface by a covalently attached glycolipid group, closes the gate in ~1 millisec. Much faster gating action (~10-8 - 10-6 sec) could be achieved by nanodevices operating variable-scale nanopores (Section 3.3.2) in response to sensor data or other control signals. Such signals could drive the insertion or retraction of diamondoid rods, wedges, or trapdoors across the channel lumen to regulate the transmission of molecules having specific sizes, shapes, and charge distributions.

Transport channels through nanodevice surfaces may also be gated electrically [392]. In contrast to the acetylcholine receptor, which is relatively nondiscriminating and allows both inorganic and organic cations to pass, the voltage-gated calcium channel has a highly discriminating mechanism with a Ca++:Na+ permeation ratio on the order of 1000:1. (The high specificity of the voltage-gated Ca++ channel is a consequence of a single-file pore mechanism involving a pair of specific Ca++ ion binding sites. Selectivity is assured if either of the two sites is occupied by Ca++, as monovalent ions do not bind strongly enough to the free site or generate sufficient electrostatic repulsion to push the first Ca++ ion through the channel [395].) Potassium channels are 100 times more permeable to K+ than to Na+, and sodium channels favor the passage of Na+ over K+ by a factor of 12. All three of these voltage-gated channels are important in the generation and conduction of neural action potentials.

A nerve impulse is an electrical signal produced by the flow of ions across the plasma membrane of a neuron. Neuron interiors have high concentrations of K+ and low concentrations of Na+. The resting potential of a neuron is -60 mV. An action potential may be generated when the membrane potential is slightly depolarized to -40 mV. This opens the Na+ voltage-gated channels, rapidly accelerating depolarization to a peak of +30 mV in ~1 millisec. Then Na+ channels close and K+ channels open, allowing K+ ions to exit the cell, restoring the -60 mV resting potential. Only ~1 ion of every ~106 Na+ and K+ ions present in the local extracellular medium and the axoplasm participate in each such nerve impulse.

The sodium channel is a single polypeptide chain with four repeating units. Each repeating unit folds into six transmembrane (alpha) helices, including one that is positively charged called the S4 helix. The S4 helix is the voltage sensor that triggers the opening of the gate. Three positively charged residues on each S4 helix are paired at the resting membrane potential with negative charges on other transmembrane helices in a staircase geometry. The initial small depolarization event produces a spiral motion of each S4 accompanied by the net movement of one or two charges to the extracellular side of the membrane, essentially turning this left-handed hydrogen-bonded "molecular screw" through a ~60° rotation [395]. This outward 0.5-nm translation of the four S4 segments opens the sodium gate by removing a steric barrier to ion flow. The energy cost of moving ~6 electrical charges (~10-18 coul) from the cytosolic to the extracellular side of the membrane against a ~100 mV potential (thus opening the gate in ~75 microsec) is ~100 zJ. Quantum tunnelling activation of sodium channels, taking 1-1000 microsec, has been analyzed by Chancey [679].

Artificial ion-gated polymer membranes were reported in 1982 [393], protein engineering of switchable pore-forming proteins is well-known [880], and "intelligent gels" are being developed that can change size and molecular porosity in response to chemical, electrical or thermal stimuli. In 1997, however, electroporation was a more commonly used method in biological research and a useful technique for "transfecting" cells in genetic studies. Electroporation employs a brief intense pulse of electricity to provide a force that opens cellular pores, enabling the insertion of macromolecules like DNA into cells of interest; laser pulses reduce cell loss to 10% by using a square-wave pulse to effect rapid and reversible pore formation [1295].

The first true voltage-gated nanomembrane was fabricated by Charles Martin and colleagues in 1995 [394]. This membrane consists of cylindrical gold nanotubules with inside diameters as small as 1.6 nm. When the tubules are positively charged, cations are excluded and only negative ions are transported through the membrane. When the membrane receives a negative voltage, only positive ions are transported through the tubules. Nanodevices may combine voltage gating with pore size and electrosteric constraints to achieve precision transport control with moderate molecular specificity at diffusion-limited throughput rates.


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

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