**Kinematic
Self-Replicating Machines**

**©
2004 Robert A. Freitas Jr. and Ralph C. Merkle. All Rights Reserved.**

Robert A. Freitas Jr., Ralph C. Merkle, *Kinematic Self-Replicating Machines*,
Landes Bioscience, Georgetown, TX, 2004.

**5.9.5 Power Law Scaling
in Convergent Assembly Nanofactory Systems**

J. Storrs Hall has made an interesting observation concerning a discrepancy in the power law scaling of convergent assembly nanofactories as compared with biological systems, and has produced a mathematical analysis [2868] from which the following discussion is drawn extensively.

As noted in Section 5.9.4, one of
the standard architectures for a molecular manufacturing system (*Nanosystems*
[208] at Section 14.3.1.b) is convergent assembly,
in which a very large array of very small manufacturing machines makes molecular-scale
parts, after which a second array of not-quite-as-small manufacturing machines
puts parts together into bigger parts, and so forth. At each stage the number
of manufacturing machines decreases by some fraction, and the size of the parts
increases by some fraction. For example, in the proposal by Merkle [213]
(Section 5.9.4) a cubical system accepts inputs from
4 half-size (1/8 volume) precursor cubical systems. The half-size cubes run
twice as fast as the big ones – the product moves through them at the
same absolute speed but only has to go half as far – and the 4 half-size
cubes feed 8 subassemblies to the full-size cube in the time required for the
full-size cube to produce one output assembly. Each smaller cube is in turn
fed by 4 smaller cubes, and so forth. Adding a new stage of double the size
increases system mass M_{system} by a factor of 8. But since the new
stage includes only four cubes of the prior stage, output has only increased
by a factor of 4, so the time (t) the new system
takes to produce its own mass in output is twice as long as the old one. This
is called a one-third power scaling law of replication time to mass –
that is, replication time is proportional to the cube root of the mass, i.e.,
t ~ M_{system}^{1/3}.

Hall [2868] shows that
any strictly self-similar convergent assembly nanofactory system in which each
stage is a scale model of the next, and where the same scale factor (and thus
the same mass ratio) applies to all stages, must also have a one-third power
scaling law. To demonstrate this, and as a generalization of the case examined
in Section 5.9.4, Hall considers a system with a branching
factor of n in units of mass of a (k)-stage system. If x is the mass of the
(k + 1)-stage node, then the mass of the full (k + 1)-stage system is (n + x)
and so the mass ratio of a (k + 1)-stage system to a (k)-stage system is (n
+ x). Using the time to replicate a k-stage system as the unit of time, the
output per unit time of a (k + 1)-stage system is n, hence the time to replicate
for a (k + 1)-stage system is t_{k+1} = (n
+ x) / n. The mass ratio of the (k + 1)-stage node to the k-stage node is also
(n + x): The ratio of the head node to the whole system is x / (n + x), which
equals the ratio of the k-stage node to the whole subsystem it is the head of;
but since that is 1 by definition, then the mass of a k-stage node is just x
/ (n + x), so the ratio of the (k + 1)-stage node to the k-stage node is x /
(x / (n + x)) = (n + x).

For a power scaling law in 1/Z of the form t
~ M_{system}^{1/Z}, the replication time raised to the power
Z equals the mass ratio, that is, t_{Z} =
((n + x) / n)^{Z} = (n + x), which, after a series of algebraic manipulations,
yields in turn: (n + x)^{(Z-1)} / n^{Z} = 1, (n + x)^{(Z-1)}
= n^{Z}, (n + x) = n^{(Z/(Z-1))}, therefore x = n^{(Z/(Z-1))}
– n, or (n + x) = n^{(Z/(Z-1))}. Because of strict scaling, the
area of the output port on any node scales with the area of a side, and since
the flow of material out of a node must equal the total flow into it, then (n
+ x)^{2/3} = n or (n + x) = n^{3/2} = n^{(Z/(Z-1))},
and Z = 3. Thus Z must equal 3 for any strictly self-similar convergent assembly
nanofactory system, i.e., such systems must have a one-third power scaling law
of replication time to mass.

The discrepancy with biology occurs because natural systems,
like animals and plants, empirically have a well-documented one-fourth power
(Z = 4) scaling law for replication time (Section 5.2),
believed to be related to the fractal nature of vascular and similar systems.
In other words, if replicator system mass increases by a factor of 1000, replication
time rises by a factor of 10 for a convergent-assembly nanofactory having Z
= 3 but only by a factor of 5.62 for biological replicators having Z = 4. Hall’s
contribution is his insight that, in principle, convergent-assembly nanofactories
can also achieve one-fourth power (Z = 4) scaling, matching biological systems,
if the length of units is allowed to scale by one factor (L) while the radius,
incorporating the other two spatial dimensions, is allowed to scale by a different
factor (R). In this case the mass ratio becomes L · R^{2} = (n
+ x); Z = 4 implies that (n + x) = n^{4/3}, fixed output area implies
that R^{2} = n, therefore L = (n + x) / R^{2} = n^{4/3}
/ n = n^{1/3}.

One simple nanofactory design following the new rules allowing
Z = 4 is to let n (the branching factor) equal 2, producing a binary tree structure
with L = 2^{1/3} and R = 2^{1/2}. The design (Figure
5.15) is a fractal plumbing embedded in a box with relative dimensions 1
x 2^{1/3} x 2^{2/3}. This box can be recursively divided in
half, by running in a pipe from the center of a large side and slicing to halve
the long dimension, giving two boxes similar to the first one. The pipes decrease
in length by a factor of L_{k-1}/L_{k} = n^{-1/3} =
2^{-1/3} = 0.7937 and decrease in radius by a factor of R_{k-1}/R_{k}
= n^{-1/2} = 2^{-1/2} = 0.7071. However, cautions Hall: “It
remains to be seen whether we can design actual assembly machinery along these
lines.”

Last updated on 1 August 2005