Nanomedicine, Volume I: Basic Capabilities
© 1999 Robert A. Freitas Jr. All Rights Reserved.
Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999
22.214.171.124 Anchoring and Dislodgement Forces
A footpad which is noncovalently bound only to a single lipid1481 or unanchored protein1482,1483 molecule embedded in the plasma membrane will slowly move as the membrane molecule(s) to which it is attached experiences translation diffusion according to the well-known Einstein-Smoluchowski equation:
which is readily obtained by combining Eqns. 3.1 and 3.5. For lipidic probes, translational diffusion coefficients in artificial fluid bilayer systems generally range from D = 1-10 x 10-12 m2/sec;1430 for plasma membrane lipids in human fibroblasts, D = 1.6 x 10-12 m2/sec at 310 K.1431 Proteins reconstituted in phospholipid bilayers show diffusion coefficients from D = 0.7-8 x 10-12 m2/sec. LDL receptors in human fibroblasts have D = 0.2 x 10-12 m2/sec at 310 K; MHC Class I (HLA) antigens in human neutrophils, lymphocytes and fibroblasts have D = 0.05-0.07 x 10-12 m2/sec, or 0.15-0.30 x 10-12 m2/sec in human endothelial cells.1482 There is a weak dependence on protein molecular size1430 -- according to Monte Carlo simulations, a plasma membrane lattice covered up to 82% with impermeable (protein) domains diminishes D by only a factor of 20 in comparison to the zero concentration limit.1436 In erythrocyte plasma membranes, spectrin-anchored proteins such as the anion transport protein band 3 have D ~ 0.0045 x 10-12 m2/sec, while D = 0.25 x 10-12 m2/sec in spectrin-depleted cells where the same proteins are no longer anchored;1432,1434 in either case, D = 0.8-1.5 x 10-12 m2/sec for RBC plasma membrane lipids.1432,1433 A few covalently glycolipid-linked proteins show exceptionally stable anchorage,1485 such as the sperm antigen PH-20 with D = 0.00001 x 10-12 m2/sec.1482
Taking t ~ nleg-1 ~ 10-100 microsec (Section 126.96.36.199) as the duration of the anchoring event during legged ambulation, then attachment to a lipid or an unanchored protein (D ~ 10-12 m2/sec) gives a footpad wander of DX ~ 4-14 nm, which is probably acceptable in most applications. Footholds on transmembrane proteins that are mechanically linked to the submembrane cytomatrix will require a force of at least 100 pN/molecule to dislodge, and will have even less free play, given the lower D. Additionally, for a cylindrical protein of radius R ~ 3 nm traversing a plasma membrane of thickness h ~ 10 nm, the rotational diffusion coefficient Drot is given by:1483
taking hmembrane ~ 10 kg/m-sec at 310 K, which gives a diffusional rotation of Da = (2 Drot t)1/2 = 0.1-0.3 radian taking t ~ 10-100 microsec.
For much longer parking times (e.g., t ~ 1 sec), in the classical fluid mosaic model of the cell plasma membrane (Section 188.8.131.52), DX ~ 1 micron for footpads attached to lipids or unanchored proteins. However, recent work by Kusumi and Sako1476 has demonstrated that a substantial fraction of unanchored proteins are transiently confined to domains somewhat smaller than this. According to their membrane-skeleton fence model, a spectrin-like meshwork (Fig. 8.43) closely apposed to the cytoplasmic face of the plasma membrane sterically confines transmembrane proteins to regions on the order of the cytoskeletal mesh size. The fences appear elastic, because, for example, transferrin receptors rebound after they strike barriers, and a small fraction of these receptors seem to be fixed to the underlying cytoskeleton by spring-like tethers.1477 For cadherins, transferrin receptors, and epidermal growth factor receptors, the domains (e.g., the barrier-free path or BFP) are 300-600 nm in diameter and confinement lasts 3-30 sec.1476-1478 BFPs for the lipid-linked and membrane-spanning isoforms of the MHC (Section 184.108.40.206) antigens are ~1700 nm and ~600 nm, respectively, at a temperature of 296 K.1479 Confinement was also found for a lipid-linked isoform of neural cell adhesion molecules (NCAMs) in muscle cells (which cannot be directly trapped by the cytoskeletal network), with BFP domains ~280 nm in diameter and a mean trapping time of ~8 sec.1480 Still, footholds to well-anchored proteins may be preferred for the longest parking times.
An annular footpad of radius Rfoot = 10 nm firmly attached around its circumference to a ring of plasma membrane-surface lipid molecules, taking each molecule of area Alipid ~ 0.4 nm2/molecule and assuming a single-lipid extraction force Flipid ~ 1 pN (Section 220.127.116.11), achieves an anchorage force Fanchor = 2 p Rfoot Flipid / Alipid1/2 ~ 100 pN (~3 atm); somewhat more compact anchor geometries are possible. Equally strong footholds may be gained using artificial amphipathic transmembrane anchors terminated with ~20-nm-diameter expansible submembrane compartments (analogous cytosolic "anchor domains" are common in cytochemistry; see Figs. 8.33 and 8.34), or using amphipathic transmembrane anchors which are directly but reversibly secured to the cytomatrix. Taking the membranolytic limit as 3 x 106 N/m2,1422 then a 100 nm2 footpad can apply up to 300 pN of anchoring force without tearing the plasma membrane.
A cluster of ten 100-pN anchors can resist at least ~1000 pN of dislodgement force applied to a medical nanorobot. Most in vivo dislodgement forces will be considerably smaller than this. For example, mean shear stress at blood vessel walls due to normal blood flow is ~2 N/m2 (Section 18.104.22.168), giving a typical dislodgement force of Fdis ~ 2 pN on a 1-micron2 nanorobot. (The net attractive force of ~70 pN between red cells in a rouleaux is also ~2 N/m2; Section 9.2.3.) Shear stress in partially occluded arteries produces a maximum Fdis ~ 40 pN, and shearing stresses between venule endothelium and a rolling leukocyte may reach 100 N/m2 (Section 22.214.171.124), giving Fdis ~ 100 pN if a white cell bumps or rolls over an anchored nanorobot. Impact by a rapidly sanguinatating nanorobot moving at 1 cm/sec (Section 126.96.36.199) produces a dislodgement force of at most Fdis ~ 200 pN.
For comparison, the adhesion strength for the protozoan Amoeba proteus has been measured as ~100-1000 nN,1456 giving an adhesion force of 100-1000 pN/micron2 over a focal contact area of ~1000 micron2.1454 The tension force exerted by a single fibroblast during locomotion has been measured as ~165 nN,1461 or ~1000 pN/micron2 (1000 N/m2). Cell-cell adhesion of T cells and target cells is ~1500 pN/micron2.1458
Cytoambulatory dislodgement forces elsewhere in the human body are equally modest. For example, taking Eqn. 9.58 and crude estimates using data from Table 9.4 and Section 8.2, shear stress is ~0.001-0.1 N/m2 along the walls of the small intestine and ~0.01-1 N/m2 in the large intestine; at the rectal wall, ~1-100 N/m2 during normal defecation and ~10-100 N/m2 during explosive defecation; on the walls of the male urethra, shear stress is ~1 N/m2 during urination and up to ~10-100 N/m2 during ejaculation; on the esophageal walls, shear stress is ~0.001 N/m2 when swallowing water, ~1 N/m2 while swallowing food, and ~10-100 N/m2 during emesis; on the corneal surface of the eye, ~0.1 N/m2 during eyelid flapping; and on the tracheal and nasal surfaces, ~0.001 N/m2 during a sneeze.
Last updated on 21 February 2003