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Derivation of ${\cal E}_{min}$ from the Capacity of Molecular Machines

The capacity of a molecular machine is given by:

$$C_y = d_{space}\log_{2}{ \left( \frac{P_y}{N_y} + 1 \right) } \;\;\;\;\; \mbox{(bits per operation)} .$$ (17)

[Schneider, 1991]. The symbols have the following meanings:

• C_y. The ``machine capacity''. Closely related to Shannon's channel capacity [Shannon, 1949], it is the maximum amount of information which a molecular machine can gain per operation.
• d<sub>space</sub>. The number of independent parameters needed to define the positions of machine parts. d<sub>space</sub> cannot be larger than 3n - 6, where n is the number of atoms in the machine.
• P<sub>y</sub>. The ``power'' or rate at which the machine dissipates energy into the surrounding environment during an operation, in joules per operation.
• N_y. The ``noise'' or thermal energy which disturbs the machine, in joules.

By dividing the power by the machine capacity at that power we obtain the number of joules that must be dissipated to gain a bit of information [Raisbeck, 1963]:

$${\cal E}\equiv \frac{P_y}{C_y} \;\;\;\;\; \mbox{(joules per bit)}.$$ (18)

Although decreasing P_y decreases ${\cal E}$, the capacity C_y also decreases according to equation (17), so we might incorrectly anticipate that at P_y = 0we would discover that ${\cal E}$ would be undefined or zero. However, ${\cal E}$ does approach a distinct limit (Fig. 1) [Raisbeck, 1963] which we can find by substituting (17) into (18):

$${\cal E}= \frac{P_y}{ d_{space}\log_{2}{ \left( \frac{P_y}{N_y} + 1 \right) }} \;\;\;\;\; \mbox{(joules per bit)} ,$$ (19)

and defining ${\cal E}_{min}$ as the limit as $P_y \rightarrow 0$(using l'Hôpital's rule [Thomas, 1968]):

$${\cal E}_{min}\equiv \lim_{P_y \rightarrow 0} {\cal E}= \frac{N_y \ln(2)}{d_{space}} \;\;\;\;\; \mbox{(joules per bit)} .$$ (20)

  

Figure: The lower bound on ${\cal E}$ is ${\cal E}_{min}$.

The thermal noise disturbing a molecular machine is:

$$N_y = d_{space}k_{\mbox{\scriptsize B}}T \;\;\;\;\;\;\mbox{(joules)}$$ (21)

[Schneider, 1991] so substituting (21) into (20) gives us

$${\cal E}_{min}= k_{\mbox{\scriptsize B}}T \ln(2) \;\;\;\;\; \mbox{(joules per bit)}$$ (22)

which is equation (16) again. The value of d_space, which is not easy to determine, conveniently drops out of the equation.

This derivation was first recognized by Pierce and Cutler [Pierce & Cutler, 1959,Raisbeck, 1963]. Because it produces the same result as equation (16), the derivation shows that the machine capacity (equation (17)) is closely related to the Second Law of Thermodynamics under isothermal conditions.

Although the present paper was written using the equations for a simple molecular machine, one also obtains equation (22) for both the Shannon receiver [Pierce & Cutler, 1959,Raisbeck, 1963] and for the general molecular receiver [Schneider, 1991] because the factors of d_space and W cancel between the capacity and noise formulas in each case. (See Table 1 in [Schneider, 1991].) So Shannon's channel capacity is, surprisingly, also related to the ``isothermal'' Second Law of Thermodynamics.