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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 p is given by the Donnan-van't Hoff
formula
| p
= (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 p ~ 1.6 x 107 N/m2
~ 160 atm. Natural bloodstream concentrations of salt produce p ~ 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 p ~
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  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  P is
applied pressure (N/m2), then avoiding boiling the
solvent water and denaturing proteins requires at least that Pmax < CVTboil, where the heat capacity of
water CV = 4.19 x 106 joules/m3-K
and Tboil = 373 K - 310 K
= 63 K give Pmax <
2600 atm. The designs presented below operate at 6% of this
maximum (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 = H
Lsieve2 / rpore2.
The volume processing rate (m3/sec) of the sieve  sieve
= chamber
= hchamber Lchamber2 / tp,
hence the molecule processing rate is  = chamber 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 = 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) = (kT trot
/ 4  rpore3)1/2
> ~2 Nrot, where is solvent viscosity (1.1 x 10-3
kg/m-sec for plasma) at T = 310 K. Solving for minimum tp
gives:
tp  32 4
Nrot2 ctarget hchamber
rpore6 / kT 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 pore
= rpore4
Psieve / 8 hsieve
and the volume processing rate through the entire sieve is sieve
= Npore pore = chamber;
solving for maximum tp gives:
tp  8
hchamber hsieve Lchamber2
/ H Psieve rpore2
Lsieve2 (sec) |
|
(3.15) |
Equating these two bracketing constraints, hsieve 150 nm for large
molecules (rpore ~ 5 nm) but hsieve 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
Dn (watts/m3) |
|
(3.16) |
where Ddevice = device power density (watts/m3),
total device power Pdevice = Pchamber + Psieve
(watts), chamber fluid flow power Pchamber = Lsieve4
Pchamber2 /
128 hchamber , sieve fluid flow power Psieve
= H Lsieve2
rpore 2 Psieve2
/ 8 hsieve , and Pchamber
= 16 H hchamber
rpore2 Psieve
/ Lsieve2
hsieve. For small molecules (e.g. NaCl or glucose), Psieve 160 atm to overcome
maximum osmotic backpressure; for large molecules (e.g. ~50,000
dalton proteins), we assume Psieve
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, Psieve = 160 atm, Pchamber = 0.0001 atm. Piston
velocity ~ 60 micron/sec and = 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, Psieve = 1 atm, Pchamber = 0.0004 atm. Piston
velocity ~ 100 micron/sec and = 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 Pclog
exceeds the thermal energy of the trapped particles, that is, if
| Pclog
> 9 kT / 8 R3 (N/m2) |
|
(3.17) |
By this criterion, Pclog
> 500 atm for small molecules and Pclog > 0.1 atm for large
molecules. From the values of 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
< 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 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.
Figure 3-3. Variable Size/Shape Apertures Using Two
Nanoscale Perforated Sliding Plates
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 - 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 - r, and the second piston pushes the remaining
filtrate back into the first chamber. After this second phase,
particles of radius ~(r ± 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 - r between the second
and third chamber, and a piston at either end (one pushing, one
pulling), thus concentrating molecules of size r ± 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 -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 -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
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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