© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

9.4.3.2.1 Plasma Membrane Areal Expansion Elasticity

A nanorobot with a footpad of width L_foot = 10 nm that applies a pressure of P_nano = 10 pN / L_foot² across a span L_foot of plasma membrane surface (and normal to it) creates an isotropic tension of T ~ P_nano L_foot = 1 x 10^-3 N/m within the membrane, causing the plasma membrane under the nanorobot to expand uniformly without shearing or bending. For comparison, the tension required to smooth out the thermal undulations or "Brownian motion" of the outer membrane of artificial phospholipid vesicles 10-20 microns in diameter (typical cell size) was determined experimentally to be 0.01-0.1 x 10^-3 N/m.³⁶⁸ At the other extreme, osmotically swollen red cells will instantaneously lyse* at a plasma membrane tension of T_lyse = 10-20 x 10^-3 N/m^1415,1421 or at an elastic modulus with respect to area dilation of T_lyse / h_cell ~ 3 x 10⁶ N/m²,¹⁴²² where red cell wall thickness h_cell = 8 nm.

* T ~ 6 x 10^-3 N/m produces red cell plasma membrane lysis after ~10 sec; time of lysis is a stochastic function that increases to ~120 days (maximum red cell lifetime) at T ~ 3-4 x 10^-3 N/m.¹⁴¹⁵

The relative area expansion of the plasma membrane under the nanorobot footpad is DA / A = T / K,¹⁴¹⁵ where the experimentally-determined area compressibility modulus is K = 0.45 N/m for red cell plasma membrane at 298 K^{1412,1413,3171,3172} (change in area compressibility modulus with temperature is 6 x 10^-3 N/m-K)³¹⁷²; K = 1.7 N/m for certain cholesterol-lipid mixtures.¹⁴¹⁴ This gives DA / A ~ 0.2% or 0.06% for the nanorobot footpad of size L_foot described above (2%-4% areal expansion produces lysis¹⁴¹⁵), or a mean linear strain of e = (DA / A)^1/2 ~ 0.04 (4%) or 0.02 (2%). For e << 1, the footpad depresses the surface by Dx ~ L_foot (e/2)^1/2 ~ 1.5 nm into the red cell's interior. This deflection represents only ~15% of lipid bilayer membrane thickness and must be considered modest in most circumstances. For leukocyte plasma membranes, measured K = 0.636 N/m.⁸⁴⁶

The design of cytoambulation mechanisms should attempt to minimize the activation of sodium-ion (and many other) stretch-activated channels -- gated transmembrane channels that are activated by simply stretching the plasma membrane, or by tension or stress development in cytoskeletal elements associated with the cell membrane.^362,1506 Such channels have already been implicated in the maintenance of cell volume.^491,1416 Biochemical transduction of mechanical strain has also been investigated in bone cells during normal loading. Linear strains of e < 0.05% were nonstimulative; those between 0.05%-0.15% maintained normal bone mass, and strains of e > 0.15% stimulated osteoblasts to increase bone mass.^1417-1419 Linear strains >1% induced osteoblasts to alter morphology, becoming fibroblast-like,¹⁴²⁰ and red cells lyse at e >~ 20%.³⁶² A limit of DA / A ~ 0.05% for a 10 nm footpad on a red cell surface would give an estimate of ~2 pN for the activation threshold. However, a force of 1 pN across a protein stretch sensor of cross-sectional area ~1 nm² represents an energy density of 10⁶ joules/m³, or 10 zJ (~2 kT) for a ~10 nm³ sensor volume, which is probably close to the limit for biological force detection consistent with earlier estimates of nanosensor detection limits (Section 4.4.1).

From these comparisons, it appears that a biological response, stimulated by plasma membrane areal expansion due to the passage of nanorobot footpads across the cell surface employing forces typical in cytoambulation (e.g., ~20 pN; Section 9.4.3.5), cannot be ruled out. However, in 1998 it was not yet known how efficiently mechanical pressures cycled at 10-100 KHz are transduced by plasma membrane stretch sensors, since most mechanical cell stimulation experiments have been conducted at frequencies between 0.05-5 Hz.* Energy coupling may be poor due to a large mechanical impedance mismatch. Looser glycocalyx anchorages also could minimize stretch activation effects.

* Specifically: 0.05 Hz,³⁵⁹⁹ 0.1 Hz,³⁶⁰⁰ 0.25 Hz,³⁶⁰¹ 0.33Hz,^3600,3602 0.5 Hz,^3603-3604 1 Hz,^3604-3609 1.67 Hz,³⁶⁰⁹ 2 Hz,³⁶⁰⁴ and 5 Hz.³⁶¹⁰

Last updated on 21 February 2003