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Derivation of the Machine Capacity

How many distinct after states can there be? Certainly the largest number of distinct states that the machine could have after dissipation of its energy cannot be bigger than the maximum number, M_y, of small after spheres that can be packed into the volume of the large before sphere, as suggested by Fig. 1 (Assumption 6, see also the second part of the definition of molecular machines) [Sloane, 1984,Conway & Sloane, 1988]. We obtain this by dividing the volume of the larger sphere by the volume of a smaller sphere [Shannon, 1949]:

$$M_y \leq \frac{ V_{before} } { V_{after} } = \left( \sqrt{ \frac{P_y + N_y}{N_y} } \; \right) ^{2 d_{space}}$$ (37)

using equations (26), (36), (35), (17), and the fact that a sphere volume is proportional to the radius raised to the dimension (D) that the sphere is embedded in. The ``machine capacity'' is the maximum information, $\log_{2}M_y$, that could be gained during the operation [Shannon, 1948,Shannon & Weaver, 1949,Pierce, 1980,Shannon, 1949]:

$$C_y = \log_{2}M_y \leq d_{space}\log_{2}{\frac{P_y+N_y}{N_y}} \;\;\;\;\;\mbox{(bits per operation).}$$ (38)

Aside from a constant due to the nature of the different situations, this equation is identical in form to Shannon's famous channel capacity formula (equation (45) in Appendix 20). In Appendix 21 we discuss how Shannon's precision theorem applies to the case of molecular machines and in Appendix 23 we discuss a more general derivation.