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Discussion

The derivation of ${\cal E}_{min}$ from the Second Law of Thermodynamics is almost certainly the one that von Neumann gave during his lectures at the University of Illinois in 1949 [von Neumann, 1966]. Ironically, his exact words were lost because of noise in a bad tape recording (equation (21)!), and he died before he could complete his book. ${\cal E}_{min}$ has been derived in other ways [Brillouin, 1962,Landauer, 1961,Keyes & Landauer, 1970,Keyes, 1970] that do not demonstrate its generality.

Equations (15), (16) and (20) are ``nothing more than'' restatements of the Second Law (equation (13)) [Jaynes, 1988,Ehrenberg, 1967]. The derivation holds not only for the machine capacity, but also for Shannon's channel capacity [Shannon, 1949] and the general molecular receivers described in [Schneider, 1991]. Thus all three theories described in the appendix of [Schneider, 1991] give the same value for ${\cal E}_{min}$. It is surprising that the close relationship between the Second Law and the channel capacity is not well recognized, since the channel capacity formula has been known since 1949.

In the general molecular receiver theory [Schneider, 1991], there are two ways for the power to approach zero to attain the limit ${\cal E}_{min}$when the temperature is held constant. Since

$$P_z = \frac{-q}{t} \;\;\;\;\;\;\mbox{(joules per second)} ,$$ (23)

one of these is to decrease the amount of energy dissipated, $-q\rightarrow 0$, while the other is to increase the amount of time, t, that the machine takes to perform its decoding operation. Thus taking the limit as $P_z \rightarrow 0$ corresponds to taking the limit as $t \rightarrow \infty$when the energy dissipation -q is held constant. Since this limit produces the isothermal Second Law, and since we all have been taught that the equality in the Second Law only holds for ``reversible'' machines, we have here a particularly neat way to see the Second Law as the limit of extremely slow ``reversible'' operations (equation (20)). The same argument holds for Shannon's theory. In contrast, simple molecular machines [Schneider, 1991] cannot take advantage of long time periods and P_y = -q, so only $P_y \rightarrow 0$ is relevant.

Because Shannon's channel capacity theorem applies to formula (18) [Shannon, 1949,Schneider, 1991], we can see that

The word ``precise'' means that so long as the bound is not exceeded, the error rate may be made as small as desired.

Another consequence of the channel capacity theorem is that even single molecules can perform precise Boolean logic if they do not exceed the machine capacity. This suggests that fast and accurate molecular computers are possible [Feynman, 1961,Drexler, 1981,Carter, 1984,Haddon & Lamola, 1985,,Conrad, 1986,Drexler, 1986,Arrhenius et al., 1986,Hong, 1986,Hopfield et al., 1988,] and that these may operate close to ${\cal E}_{min}$. Although we do not know how to design them yet, computers built from proteins are well within our present construction capabilities [Maniatis et al., 1982,Beaucage & Caruthers, 1981,Pabo, 1983,,Rastetter, 1983,Wetzel, 1986,Lesk, 1988].

I thank Herb Schneider and John Spouge for enormously fun and useful discussions, John Skilling for a supportive letter, Peter Basser, Peter Lemkin, Sarah Lesher, Joe Mack, Jake Maizel, Peter Rogan, Denise Rubens and Morton Schultz for critically reading the manuscript and discussing these ideas. I also thank Gary Stormo for pointing out that base pairing requires a 1 in 4 choice, and Larry Gold for supporting the preliminary stages of this project under NIH grant GM28755.