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Nanomedicine

by Robert A. Freitas Jr.

© Copyright 1998, Robert A. Freitas Jr.
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Chapter 3. Molecular Transport and Sortation

 

3.4 Receptor-Based Transport

The most efficient of all modes of molecular transport involves receptor sites capable of recognizing and selectively binding specific molecular species. Many receptors reliably bind only a single molecular type; others, such as sugar transporters, can recognize and transport several related sugar molecule types. In nanomechanical systems, artificial binding sites of almost arbitrary size, shape, and electronic charge may be created and employed in the construction of a variety of highly efficient molecular sortation and transport devices, described below. A discussion of binding sites and receptor engineering follows in Section 3.5.

 

3.4.1 Transporter Pumps

While gated channels enable ions to flow rapidly through membranes in a thermodynamically downhill direction, active pumps use a source of free energy to force an uphill transport of ions or molecules. In biology the energy supply is usually ATP or photons of light; medical nanomachines can make use of vastly more diverse energy sources (Chapter 6). Actually, the term "pump" may be something of a misnomer because the action is highly specific -- only one or a very small number of molecular species are selectively transported.

Molecular pumps generally operate in a four-phase sequence: (1) recognition (and binding) by the transporter of the target molecule from a variety of molecules presented to the pump in the input substrate; (2) translocation of the target molecule through the membrane, inside the transporter mechanism; (3) release of the molecule by the transporter mechanism; and (4) return of the transporter to its original condition, so that it is ready to accept another target molecule. Such molecular transporters that rely on protein conformational changes are ubiquitous in biological systems, and are illustrated schematically in Figure 3-5.

 

Transporter Molecular Pump

Figure 3-5. Schematic of Transporter Molecular Pump Operation (Uniport)

The minimum energy required to pump molecules is the change in free energy (delta)G in transporting the species from one environment having concentration c1 to a second environment having concentration c2, given by

(delta)G = kT ln(c2/c1) + Ze F(delta)V / NA   (3.18)

where k = 0.01381 zJ/K (Boltzmann constant), T = 310 K, Ze is the number of charges per molecule transported (i.e. the valency), F = 9.65 x 104 coul/mole (Faraday constant), (delta)V is the potential in volts across the membrane, and NA is Avogadro's number. So for example, transport of an uncharged molecule across a c2/c1 = 103 gradient (typical in biology) costs ~30 zJ. An extremely aggressive c2/c1 = 106 concentration gradient costs ~60 zJ/molecule, plus another ~30 zJ/ion if we are moving Ca++ ions against a 100 mV potential. An artificial nanopump of dimension ~10 nm moving at a conservative ~1 cm/sec velocity operates at MHz frequencies, transporting ~106 molecules/sec for a continuous power consumption of ~0.03 picowatts at c2/c1 = 103. Such a pump should mass ~10-21 kg.

Transporter pumps need not be limited to the movement of a single molecular species in a single direction, which biochemists call a uniport transport mechanism. Numerous well-known biological systems are capable of moving two molecules simultaneously in one direction (symport mechanisms), two molecules sequentially in opposite directions (antiport mechanisms), and charged molecules in one direction only, thus building up an electrical charge on one side of the membrane (electrogenic mechanisms). Such pumps exist in nature for numerous ions, amino acids, sugars, and other small biomolecules [398]. Active drug efflux systems [400] and multidrug resistance [399] are made possible by the expression of bacterial genes coding for molecular pumps that are constantly evolving new specificities to increasing numbers of microbicidal drugs.

The action of the Na+ - K+ antiporter, a familiar ion pump present in all mammalian cells, is illustrated in Figure 3-6. Three Na+ and two K+ ions are transported per 10 millisec cycle, requiring the hydrolysis of one ATP molecule to ADP to drive the conformational changes. (More than one-third of the ATP consumed by a resting animal is used to pump these two ions.) Hydrolysis of ATP liberates ~80 zJ/molecule of free energy, so the antiporter is transporting Na+ and K+ at a cost of 16 zJ/ion at a 0.5 KHz frequency. Pump site density is ~1000/micron2 of cell membrane in neural C fibers [800]. (The Na+ - K+ pump can also be operated in reverse to synthesize ATP from ADP by exposing the mechanism to steep ionic gradients.) By contrast, artificial nanomechanical antiporter and symporter devices will operate at MHz frequencies [1177]. They should be able to transport much larger molecules, and may also be fully reversible.

 

Sodium-Potassium Ion Pump

Figure 3-6. Schematic of Sodium-Potassium Antiporter Ion Pump Operation

 

3.4.2 Sorting Rotors

Drexler's molecular sorting rotor [10] is a related class of nanomechanical device capable of selectively binding molecules from solution and then transporting these bound molecules against concentration gradients (Figure 3-7). The archetypal sorting rotor is a disk with 12 binding site "pockets" along the rim exposed alternately to the external solution and interior chamber by axial rotation of the disk. (Other designs may have more, or fewer, pockets.) Each pocket selectively binds a specific molecule when exposed to the solution. Once the binding site (Section 3.5) rotates to expose it to the interior chamber, the bound molecules are forcibly ejected by rods thrust outward by the cam surface (or using some other means by which receptor affinity can be adjusted during the inbound transport process). In the case of protein molecules, the debinding geometry must be carefully designed to avoid denaturation during ejection. Also, the rotor in Figure 3-7 implicitly assumes that target molecules remain in the liquid or gaseous state after importation. M. Krummenacker observes that most bloodborne molecular species will precipitate as solids unless they are well-solvated; thus nanomedical sorting systems may require internal solvent or in some cases should be made completely eutactic (Section 3.4.3). The discovery of positionally disordered water molecules resident inside protein hydrophobic cavities [1047] suggests that good rotor designs may also need to include solvent drainage channels.

 

Molecular Sorting Rotor

Figure 3-7. Molecular Sorting Rotor (modified from Drexler [10])

Molecular sorting rotors can be designed from about 105 atoms (including housing and pro rata share of the drive system), measuring roughly 7 nm x 14 nm x 14 nm in size with a mass of 2 x 10-21 kg. Rotors turn at ~86,000 rev/sec with a conservative rim speed of 2.7 mm/sec and an almost negligible drag power of ~10-16 watts against the fluid, sorting small molecules at a rate of 106 molecules/sec with laminar flow. From Eqn. 3.18, the energy cost of small-molecule sortation at 310 K ranges from ~10 zJ/molecule at low pressures (c2/c1 = 10) up to ~40 zJ/molecule when pumping against the highest head pressures (c2/c1 = 104, ~30,000 atm for natural bloodstream concentrations of salt with osmotic ppi ~3 atm), consuming 0.01-0.04 picowatts per device in continuous operation. Rotors are fully reversible, so they can be used to load or unload target molecules depending on the direction of rotor rotation. Cylindrical rotors with many receptor rows are somewhat more energy-efficient, and rotor lifetimes should be >106 sec (Chapter 13).

Typical molecular concentrations in the blood for target molecules of nanomedical interest are ~10-11 - 10-3 molecules/nm3, which should be sufficient to ensure ~99% occupancy of rotor binding sites (Section 3.5.2). Rotors targeting serum hormones and other low-concentration species at the parts-per-billion level must slow to <1 rev/sec to ensure complete receptor occupancy and to avoid exceeding diffusion limits.

Drexler [10] has also proposed cascades of sorting rotors (Figure 3-8) to achieve high fidelity purification and a contaminant fraction of <10-15. However, since only ~1010 small molecules can be stored in 1 micron3 volume (typical for nanomedical devices), 100% process purity requires a contaminant fraction of only <10-10 which can be ensured in micron-scale nanosystems using at most 5 stages, starting from a dilute input substrate containing only 1 part per billion of the target molecule with each stage providing a concentration factor of ~104. For statistically pure extractions of more common molecules in blood and cytoplasm, a 3- or 4-stage cascade will usually suffice. Note that the optimal receptor structure may differ at different stages in a cascade [10], and that each 12-arm outbound rotor can contain binding sites for 12 different impurity molecules. The first-stage receptor will likely pass only a relatively small number of different contaminant species, so the number of outbound rotors in the entire system can probably be reduced to a small fraction of the number of inbound rotors.

 

Sorting Rotor Cascade

Figure 3-8. Sorting Rotor Cascade (modified from Drexler [10])

 

3.4.3 Internal Transport Streams

After pump mechanisms described above have reliably sorted externally-encountered target molecules into reservoirs filled with species of a single type, a medical nanodevice may require these molecules to be transported to specific internal locations for further processing. Bulk fluid flow or fluidized (solvated or suspended) transport through nanopipes may suffice for some purposes (Section 9.2.5). However, in many cases it will be necessary to present reagent molecules to other subsystems as a well-ordered stream of precisely positioned moieties transported in vacuo, especially for mechanochemical operations (Chapter 19).

For this purpose, Drexler [10] proposes molecular mills -- eutactic systems of nanoscale belts moving over rollers, with reagent-binding devices mounted on the belt surface (Figure 3-9). This class of device can be assembled into complex molecular transportation networks using conditional switching, crossed-axis belting, and transit speed/frequency multipliers [10], and may also be employed to drive mechanosynthetic chemical reactions. The benchmark mechanism uses 10-nm diameter rollers to carry closely packed reagent devices measuring 4 nm x 4 nm x 2 nm, or 32 nm3. A 20-roller mill mechanism 1 micron long has a ~2 micron long belt with 500 reagent devices and delivers 106 molecules/sec at a belt speed of 4 mm/sec. Total power dissipation is ~1.4 x 10-18 watts, a rate of ~0.001 zJ per moiety (or per reagent device) delivered or ~10-6 zJ/nm traveled per reagent device. Total mill mechanism mass is ~6 x 10-20 kg.

 

Molecular Mills

Figure 3-9. Molecular Mills for Internal Transport: Simple Transfer Between Two Reservoirs

An alternative to roller/belt mill mechanisms is a non-connected stream of pallets pushed along tracks, also in vacuo. Such tracks may include merging junctions, distribution junctions, multi-plane crossings and switching stations, as well as straight and curved sections. Assuming each pallet is a 32 nm3 reagent device held to the track by pins in grooves resembling cam followers, energy dissipation by phonon scattering [10] is given approximately by:

Pdrag = 4/3 epsilonp sigmatherm v2/vsound   (3.19)

where epsilonp = 2 x 108 joules/m3 (phonon energy density), sigmatherm (a thermally-weighted scattering cross section) ~10-20 m2 for reagent devices of mass m = 10-22 kg assuming a sliding contact of stiffness ~30 N/m in a moderately stiff medium, v = 4 mm/sec sliding speed, and vsound = 104 m/sec (~speed of sound in diamond), giving Pdrag ~4 x 10-21 watts per reagent device, or Pdrag / v ~ 10-6 zJ/nm traveled per reagent device (pallet). Note that volume containerization of pallet-transported molecules is least efficient at the smallest scales, where surface area per unit enclosed volume is highest, since energy usage is proportional to the surface area of the carrier. Containerization of n>>1 molecules for large-pallet transport is more efficient.

A less energy-efficient, but far more versatile, internal molecular transport device is the 100-nm telescoping manipulator arm [10] described in Section 9.3.1.4. This flexible ~10-19 kg device employs a binding tip to pick and place small and large molecules alike, moving them at ~1 cm/sec with repeatable placement accuracy of 0.04 nm. Multiple devices can be used to establish an internal ciliary transport system (Section 9.3.4); standardized volume containerization of molecules permits rapid stereotypical handoff motions and efficient parcel routing. Conveyance through a 100-nm arc takes 10-5 sec consuming 0.1 picowatt while the arm is in motion, or ~10 zJ/nm traveled per reagent molecule or per container transported (vs. ~1000 zJ per typical covalent bond).

In molecular cytobiology, vesicles and organelles are transported throughout the interior of a cell by riding on microtubular cables crisscrossing the cytosol (Section 8.5.3.11). For example, neural vesicles show transport speeds up to 2-4 microns/sec [938].

 

3.5 Molecular Receptor Engineering

Molecular recognition requires a detailed surface complementarity between the target molecule and its receptor. The interplay of various molecular forces between ligand and receptor cause them to selectively bind together, typically engineered to occur in ~10-6 sec. It is useful first to briefly review and quantify the principal physical forces at work.

 

3.5.1 Physical Forces in Molecular Recognition

Covalent bonds, which occur when atoms share electrons, are the strongest bonds. Aside from metals and salts, most material objects are made of atoms held together by covalent bonds. The atoms comprising biological molecules like proteins, nucleic acids and lipids are strung together mostly by single, double, or triple covalent bonds, as are receptors and diamondoid nanomechanical structures. Interatomic bond strengths range from 181 zJ/bond for O-F up to 1785 zJ/bond for C-O [763]. Covalent bond lengths range from 0.10-0.16 nm within CHON-atom molecules, giving a typical covalent bond rupture force of ~10 nN/bond.

But as Jean-Marie Lehn points out, "there is a chemistry beyond the molecule" -- noncovalent supramolecular chemistry [765]. The bonds employed in molecular recognition are weak noncovalent bonds. Noncovalent bonds are largely responsible for the secondary and higher order structure of macromolecules. On a per-bond basis, noncovalent bonds are 1-3 orders of magnitude weaker than covalent bonds. However, the possibility of combining within a limited area a great number of noncovalent bonds having complementary elements allows the formation of a large specific association whose affinity may be of the same order of magnitude as a covalent bond [401]. The high combinatorial diversity provided by many complementary elements allows numerous orthogonal specific associations, enabling self-assembly of many components; by comparison, covalent chemistry offers a poor diversity of reactivities. An additional advantage is that the formation of noncovalent bonds often is not hindered by high energy barriers. At least five types of noncovalent bonds may be distinguished.

First is the electrostatic bond between two charged particles (e.g. the "salt bridge" in proteins), a dipole interaction whose energy Ee is given by Coulomb's law as:

Ee = (e2 / 4 piepsilon0)(Z1 Z2/(kappa)e r) exp(-Kdh r) (joules) (3.20)

where e = 1.60 x 10-19 coul (elementary charge), epsilon0 = 8.85 x 10-12 farad/m (permittivity constant), Z1 and Z2 are the numbers of attractant charges, r is the distance between the charges, and (kappa)e is the dielectric constant (74.3 for pure water at 310 K, usually reduced to ~40 in a hydrophobic environment). The bond is strengthened if the charges are in a hydrophobic environment. Conversely, the presence of electrolytes weakens the bond energy due to a shielding effect, given by Kdh, the Debye-Huckel reciprocal length parameter, which has a value of 1.25 nm-1 for 0.15 M NaCl (~1% solution, ~human blood). Thus two unit charges separated by 0.3 nm produce an interaction energy of Ee = 19 zJ in a hydrophobic environment, 10 zJ in pure water, and 6.3 zJ in 1% salt water.

Most isolated amino acids in neutral solution are zwitterionic -- the molecule has no overall charge but carries both a negatively charged group (carboxyl, CO2-) and a positively charged group (amino, NH3+). In proteins the individual amino acids are polymerized, giving a peptide backbone which is electrically neutral except for the ends of the chain. Most of the standard amino acids found in proteins have uncharged side chains, although histidine, lysine and arginine each have a positive charge at neutral pH and both glutamic and aspartic acids normally carry a negative charge.

A second important noncovalent interaction is the hydrogen bond, a dipole formed when a hydrogen atom covalently bonded to an electronegative atom is shared with a second electronegative atom (typically an oxygen, nitrogen or fluorine atom), such that the proton may be approached very closely by an unshared pair of electrons. Hydrogen bonds are largely responsible for the unusual thermodynamic properties of water and ice, and the DNA double-helical and protein alpha-helical and beta-structure conformations are extensively hydrogen bonded. Highest bonding energies occur when donor and acceptor atoms are 0.26-0.31 nm apart. Typical hydrogen bond strengths in proteins are 7-50 zJ.

A third important noncovalent force is van der Waals interactions (London dispersion forces) [1149]. There is an attractive component due to the induction of complementary partial charges or dipoles in the electron density of adjacent atoms when the electron orbitals of two atoms approach to a close distance. There is also a strongly repulsive component at shorter distances, when the electron orbitals of the adjacent atoms begin to overlap, commonly called steric hindrance. The van der Waals attractive bonding energy between two parallel plates of area A and separation zsep is approximated by:

Evdw = HA/12 pizsep2   (3.21)

where the Hamaker constant H = 37 zJ for water, 66 zJ for glycerol, 340 zJ for diamond (Table 9-1). For A = 0.4 nm2 (small molecule), zsep = 0.3 nm, then Evdw = 4 zJ (water) to 8 zJ (small organic molecules) -- close to the mean energy of a thermally excited harmonic oscillator, kT ~ 4.3 zJ at 310 K. While van der Waals bonds are individually very weak, they are also very numerous since they involve all pairs of neighboring atoms. For example, experimental analysis of an antigen molecule trapped in the anti-hen egg-white lysozyme monoclonal antibody Fv active binding site found 86 distinct interatomic contact points with antigen-antibody separations ranging from 0.25-0.46 nm, averaging 0.36 nm [416]. Since intermolecular dispersion forces act on all molecules, there are probably no ligands with MW > 400 daltons which cannot be receptored (Section 3.5.5). That is, van der Waals interactions ensure that virtually all molecules of nanomedical interest are theoretically bindable noncovalently.

A fourth type of interaction (pi electron to pi electron), called "aromatic" or "pi" bonding, occurs when two aromatic rings (conjugated pi systems) approach each other with the plane of their aromatic rings overlapping, with successive pi-bonded systems stacked like layers in a cake. This results in a noncovalent attractive force with a bond strength of ~40-50 zJ. pi bond stacking forces contribute to nucleic acid stability at least as much as the hydrogen bonds between the bases [401].

Fifth, there are the strong hydrophobic forces, due entirely to solvent entropy changes. When two nonpolar residues approach each other, the surface area exposed to solvent is reduced, increasing the entropy of all the water present and decreasing the entropy of the residues, adding to the binding energy a hydrophobic free energy of ~17 zJ/nm2 of contact surface area that was formerly exposed to water [413].

In designing an artificial binding site, the above forces may be combined to achieve the desired level of affinity and specificity for a given ligand. All forces are not equally useful in this regard, however. For example, hydrophobicity is the major factor in stabilizing protein-protein associations [402]. But hydrophobicity is almost entirely nonspecific, hence contributes little to ligand discrimination. By contrast, the proper formation of hydrogen bonds and van der Waals contacts require complementarity of the surfaces involved. Such surfaces must be able to pack closely together, creating many contact points, and charged atoms must be properly positioned to make electrostatic bonds. Thus van der Waals and polar interactions may contribute little to the dynamic stability of the ligand-receptor complex, but they do determine which molecular structures may recognize each other [402]. Other design elements of binding sites, such as directed channeling of substrates into the receptor, may also prove useful.

In analyzing molecular forces, note that at the nanoscale level, surface/surface, molecule/surface, and molecule/molecule interactions may feature very complicated behaviors. Nanodevices performing work may generate both thermodynamic and mechanical local nonequilibrium conditions, so calculations based on the general forms of interactions and on macroscopic expressions valid at equilibrium conditions should be taken only as basic estimates.

 

3.5.2 Ligand-Receptor Affinity

For a ligand binding to a receptor in a solvent, there will be a characteristic frequency with which existing ligand-receptor complexes dissociate as a result of thermal excitation, and a characteristic frequency with which empty receptors bind ligands as a result of Brownian encounters, forming new complexes, with the frequency of binding proportional to the concentration of the ligand in solution, cligand (molecules/nm3) [10]. For simple processes, the equilibrium constant Kd, taken in the direction of dissociation, is:

Kd = kd / ka (molecules/nm3) (3.22)

where kd is the dissociation rate constant (sec-1) and ka is the association rate constant (nm3/molecule-sec). The ka rate constant reflects mainly the molecular weight of the ligand, and thus varies little among antibody, enzyme, or other receptor systems. For example, Delaage [401] notes that changing the solution pH for growth hormone from 7.5 to 4.0 increases Kd by a factor of 3000, due to kd increasing by a factor of 1600 but ka decreasing only by a factor of 1.7. Hence it is the rate constant of dissociation, kd, which accounts for the vast bulk of affinity in receptor systems.

Thus receptor affinity is usually taken as the inverse of the dissociation rate constant, which may be placed in the context of the half-life of the ligand-receptor complex approximated [401] by:

t1/2 ~ ln(2) / kd (sec) (3.23)

Observed half-lives range from <0.1 microsec (kd ~ 107 sec-1) for the enzyme catalase to a few months (kd ~ 10-7 sec-1) for enzyme inhibitors such as the Kunitz inhibitor of trypsin [406] and for avidin-biotin binding [407]. The smaller the kd (or the Kd), the greater the affinity and so the more firmly the receptor grasps the ligand.

The probability Poccupied that a receptor will be occupied [10] is given by:

Poccupied = (cligand / Kd) Punoccupied   (3.24)

where Punoccupied = 1 - Poccupied. To ensure Poccupied = 99% receptor occupancy, Kd must ~ cligand / 100. For target molecules present at the 10-3 - 10-11 gm/cm3 concentrations typically found in human blood (Appendix B), cligand = 3 x 10-3 molecules/nm3 for glucose to cligand ~ 10-11 molecules/nm3 for female serum testosterone, giving a range of Kd ~ 10-4 - 10-13 molecules/nm3 to achieve 99% occupancy.

How much binding energy per receptor will this require? The free energy of dissociation (delta)Gd of a ligand-receptor complex is related to its equilibrium dissociation constant Kd by:

(delta)Gd = - kT ln(Kd / K0)   (3.25)

which refers to a standard reference state where all chemical species are 1 M (i.e. K0 ~ 0.6 molecules/nm3) and attributes a free energy of zero to a complex with a dissociation constant of 1 M [402].

For T = 310 K, the range of required Kd gives a range for (delta)Gd of 39.4 zJ for glucose (at typical serum concentrations) to 128 zJ for female serum testosterone.

However, when ligand and receptor associate there is a loss of three degrees of freedom in each of translational and rotational entropy, which may be estimated using the classical Sackur-Tetrode equations, giving an entropic free energy range (for translation and rotation combined) of (delta)Gs = 80 zJ for very small molecules (MW ~ 10 daltons), to 120 zJ (MW ~ 102 daltons), 200 zJ (MW ~ 104 daltons), and 280 zJ for large molecules (MW ~ 106 daltons) [427].

Thus to form a ligand-receptor complex with a dissociation constant Kd, the receptor design must provide a free energy of binding of at least:

(delta)Gtotal = (delta)Gd + (delta)Gs   (3.26)

or 120-410 zJ/molecule for designed receptors achieving 99% occupancy operating over the likely range of physiological concentrations and temperatures. This is consistent with Drexler's estimate of 161 zJ binding energy required to ensure reliable receptor occupancy for small plentiful molecules [10].

 

3.5.3 Ligand-Receptor Specificity

While affinity measures the strength of the binding of a ligand to a receptor, specificity defines the degree to which a receptor can distinguish between similar ligands. That is, the affinity of the target molecule for the receptor must be greater than the affinity of any other ligand in the environment that is competing for that same receptor, by some threshold multiple.

How much greater is enough? In natural dynamic cellular systems, the threshold multiple appears to be a factor of ~102-103. For example, the carrier which expels Ca++ from erythrocytes presents a variation in Kd of 10-6 to 10-3 in going from the interior to the exterior of the cell [404]. The active transport of amino acids by hepatocytes normally involves Kd ~10-1, but under conditions of deprivation a high affinity carrier with Kd ~10-3 comes into play [405].

However, the key to assessing specificity in the nanomedical context would be to tally all the competing molecules in the in vivo environment, determine which are the nearest competitors, and then design to avoid them by imposing appropriate energy barriers. By 1997, only a very few competitive ligand-receptor binding analyses had been performed [1078]. Until a complete molecular inventory of the human body becomes available, the following crude estimate of nearest-neighbor differences must suffice.

The human body contains a minimum of Nprot ~ 105 distinguishable proteins (Section 3.1). The maximum number of distinguishable proteins in the biosphere was given by Kauffman [766] who estimates a useful biological catalytic task space of Nprot ~ 108 distinct protein forms, a tiny subset of the ~20500 possible 500-residue protein sequences. If the average protein is constructed from Nresidue ~ 500 amino acids (MW ~ 50,000 daltons), then the average protein may differ from its most similar neighbor by nvar ~ log (Nprot) / log (Nresidue) = 1.9-3.0 residues. (The precise magnitudes of Nprot and Nresidue are not crucial to our conclusion.) Most proteins are confined to cells containing Nprot ~ 5000 different protein types each (Table 3-2); given that evolution has probably optimized local specificity to ensure that closely competing crucial ligands rarely appear in the same cell, it seems reasonable to assume that the average closest-neighbor ligand may differ from the average target ligand by at least nvar ~ 1 residue.

How much is receptor affinity reduced when binding molecules differing by nvar ~ 1 residue from the target ligand on receptor-accessible surfaces -- the minimum threshold required to ensure specificity within a cell? In one experiment the relative affinity of an antibody constructed for succinylglycinamide-linked histamine (histamine-SGA), which was the target molecule, and the same molecule but with one methyl group or one carboxylic group removed (and replaced with a hydrogen) was 1.45 x 104 or 2.5 x 105, respectively, due to steric hindrance [401]. Similar investigations with antibodies for SGA-linked serotonin produced relative affinities of 500-1000 [408], and with antibodies for single alanine substitutions in Human Growth Hormone (HGH), ~1000 [418]. The relative affinity of a particular RNA oligomer for theophylline and caffeine, two ligands which differ by only a single methyl group, was measured experimentally as 10,900 [1078]. Computational receptor experiments in which CH replaces N suggest a decline in relative affinity of ~5 x 104 [10]. Indeed, the change of a single hydrogen atom on a ligand is usually sufficient to destroy its specificity for (or activity within) a particular enzyme. The single-residue affinity reduction at a receptor-accessible surface appears to be of order ~103 - 105.

Each increase of ~10 zJ in bonding energy causes the reaction equilibrium constant to decline (hence receptor affinity to rise) by a factor of ~10 (Eqn. 3.25). If a difference in affinities of ~103-105 between the target molecule and its nearest competitor likely to be present in the environment provides sufficient receptor specificity for nanomedical purposes, this requirement corresponds to a binding energy differential affinity of ~30-50 zJ at 310 K between target and closest-neighbor ligands.

 

3.5.4 Ligand-Receptor Dynamics

Diamondoid structures can exhibit a stiffness and rigidity 1-2 orders of magnitude greater than that available in protein structures. In general, stiffer structures permit greater specificity because they enhance exclusion of non-target ligands based on van der Waals overlap forces (steric hindrance) and allow narrower tolerances in distinguishing acceptable ligands.

In less-stiff protein-based receptors, each of the atoms is engaged in relatively large, rapid jiggling movements. Experimental and theoretical work has been done on the atomic fluctuations within the basic pancreatic trypsin molecule, a small enzyme with 58 amino acids and 454 heavy (non-hydrogen) atoms. This work established that fluctuations increase with distance from the center of the molecule, with the magnitude of RMS fluctuations ranging from ~0.04 nm for backbone atoms to ~0.15 nm for the ends of long side chains (roughly one atomic diameter), and an average of 0.069-0.076 nm per atom over the entire molecule [409]. A similar experimental analysis of reduced cytochrome c, a common metabolic enzyme, shows that RMS fluctuations of each of the 103 amino acid residues in the molecule averages ~0.11 nm with lattice disorder (~0.05 nm) included, and fluctuations range from 0.09-0.16 nm (Figure 3-10). Antibody core domain movements display RMS fluctuations of 0.04-0.19 nm [412]. Hence it appears that the average atom within the typical protein receptor oscillates ~0.1 nm every ~10-12 sec, although frequently residues with long side chains (e.g. arg, lys) have much higher RMS deviations than average.

 

Exp. RMS Fluctuations

Figure 3-10. Experimental RMS Fluctuations in Ferrocytochrome C
(residue averages shown as function of residue number;
redrawn from Karplus and McCammon [409])

By contrast, in stiff diamondoid-based receptors each of the atoms is locked in a rigid crystalline structure and thus is subject to thermal displacements approximately 10 times smaller. The RMS displacement for a quantum mechanical harmonic oscillator [10] is given by:

(delta)X = {(hbaromega/ks)(1/2 + 1/[exp(hbaromega/kT)-1])}1/2   (3.27)

where hbar = 1.055 x 10-34 joule-sec, kT = 4.28 zJ at T=310 K, and angular frequency omega = (ksred)1/2 rad/sec where ks is mechanical stiffness and µred is the reduced mass = m1m2/(m1 + m2). For C-C atoms (e.g. in the receptor body), ks = 440 N/m, m1 = m2 = 2 x 10-26 kg, thus omega = 2.1 x 1014 rad/sec, so the RMS displacement of each atom is only ~0.005 nm every ~3 x 10-14 sec. For C-H atoms (e.g. on the hydrogen passivated receptor surface), ks = 460 N/m, m1 = 2 x 10-26 kg (C), and m2 = 1.671 x 10-27 kg (H), thus omega = 5.5 x 1014 rad/sec, so the RMS displacement of each atom is ~0.008 nm every ~1 x 10-14 sec. Similarly, at 310 K the RMS thermal displacement of a 1-nm wide, 10-nm long diamondoid rod is ~0.01 nm, including elastic and entropic contributions [10].

The ratio of RMS displacements for protein/diamondoid receptors ~10:1, so the minimum addressable volume (hence inverse maximum specificity) of a diamondoid receptor should be ~103 smaller than for protein receptors, a ~30 zJ binding energy advantage for diamondoid receptors.

 

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

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