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

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

7.2.1.3 Instantaneous Stationary Source in Stationary Medium

Consider a chemomessaging nanorobot that releases Q_message messenger molecules (each carrying I_message bits/molecule) in a single puff as a point source at time t = 0. This is the ideal design for an alarm-type message or for messages requiring rapid fadeout. For simplicity, the human body is taken to be a continuous, isotropic, unbounded, stationary aqueous medium; a more detailed treatment is beyond the scope of this book. The spatial density of message molecules as a function of time t and distance r from the point source is

{Eqn. 7.3}

where D = the translational diffusion coefficient for message molecules, which are assumed to be roughly spherically packed, estimated from Eqns. 3.5 and 7.2 as

{Eqn. 7.4}

For I_message = 100 bits, D = 2.2 x 10^-11 m²/sec; for I_message = 10⁹ bits, D ~ 1.0 x 10^-13 m²/sec. Because the detection of molecules by receptor-based chemical sensors requires a concentration-dependent minimum sensor cycle time t_EQ approximated by Eqn. 4.3, then there exists some minimum threshold concentration (c_min) of message molecules that can be detected by a chemical sensor in some minimum waiting time t_sensor = t_EQ, given by

{Eqn. 7.5}

For N_encounters ~ 100,¹⁰ N_A = 6.023 x 10²³ molecules/mole (Avogadro's number), r_message from Eqn. 7.2 and taking t_sensor = 1 sec, then c_min ~ 1 x 10^-9 molecules/nm³ for I_message = 100 bits and c_min ~ 9 x 10^-11 molecules/nm³ for I_message = 10⁹ bits. The concentration of message molecules exceeds c_min within an expanding diffusive sphere. In a time t_rec = (0.0293 / D) (Q_message / c_min)^2/3 this expanding sphere reaches a maximum size R_max = (0.419) (Q_message / c_min)^1/3, after which it begins to contract as the puff dissipates;⁷⁰³ the concentration eventually falls below c_min everywhere at t_fadeout = e t_rec, where e = 2.718...

10¹² nanorobots uniformly distributed throughout a 0.1 m³ human body have a mean interdevice separation of ~50 microns. For simple messages (I_message = 100 bits) and R_max = 50 microns, then Q_message ~ 2 x 10⁶ message molecules emitted, t_rec ~ 20 sec ('I ~ 5 bits/sec), and t_fadeout ~ 50 sec. If R_max = 1 mm, then Q_message = 2 x 10¹⁰ molecules and t_rec ~ 8000 sec to receive the signal ('I ~ 0.01 bits/sec). For complex messages (I_message = 10⁹ bits), Q_message ~ 1 x 10⁵ molecules but t_rec ~ 4000 sec (~1 hour) for the message to be transported R_max = 50 microns ('I ~ 3 x 10⁵ bits/sec).

A simple alarm signal (I_message = 100 bits) released within an individual tissue cell is received everywhere throughout the cytosol -- an assumed ~(20 micron)³ cell volume -- in t_rec ~ 0.8 sec ('I ~ 100 bits/sec) with t_fadeout ~ 2 sec, using a single alarm puff of Q_message ~ 17,000 message molecules (computed with R_max ~ 10 microns) which molecules may be stored in a (40 nm)³ volume per puff. As a practical matter, the cytosol is quite crowded with macromolecules and cytoskeletal components (Section 8.5.3); exclusion effects will increase diffusion times by as yet unknown amounts, to be determined experimentally.

Last updated on 18 February 2003