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Appendix 2: Correspondences between Molecular Machines and Communication Channels

Although the molecular machine and communication channel models are not identical, we may draw several analogies as discussed in the introduction. A molecular machine corresponds to the receiver in Shannon's theory [Shannon, 1949] since both gain energy and dissipate it to settle into a specific substate. For most molecular machines, such as the restriction enzymes on DNA, there is no transmitter, nor is there a communication channel. Rather, the formation of correct matches between molecular surfaces usually serves the function of directing the molecular machine to one or another substate. The phrase ``signal-to-noise ratio'' is not meaningful in the context of simple molecular machines.

The reader may have noticed that for channel capacity, bits were defined as a selection amongst possible symbols, whereas for machine capacity they were defined for selection among states. von Neumann [von Neumann, 1963] pointed out that so long as we can correlate events (or symbols) with states, these definitions are functionally identical.

Shannon's theory took advantage of the fact that the square of the voltage across a resistor is proportional to the power through the resistor. Likewise, the square of each y_jis the energy in the sine or cosine component of a ``pin''. Rather than using these mechanical Fourier ``potentials'', Shannon used voltage potentials in his theory. The mathematical equivalences between mechanical and electrical models are well known [Feynman et al., 1963].

The proof of the channel capacity theorem depends entirely on geometry and not on the system being modeled. However, it is important to show that the geometry applies to molecular machines. In Shannon's Figure 5 [Shannon, 1949], which is reproduced here as Fig. 10, the outer circle, with radius $\sqrt{P_y + N_y}$, corresponds to the molecular machine's spherical before state. The ``received signal'' (point A) represents only one of the possible before configurations. A spherical noise cloud around the ``transmitted signal'' (point B) of radius $\sqrt{N_y}$corresponds to an after state. Having received signal point A, the receiver must select the transmitted point B. This corresponds to a machine operation in which the sphere center moves from point O to B, as the radius collapses. Since most of the dimensions are orthogonal to OB, very little noise power extends in the direction OB, and the after sphere essentially remains inside the before sphere. The shaded region L in Shannon's figure contains centers of small spheres that have the same after configuration at A. Shannon's theorem 2 shows that the probability of having a second after sphere centered in L--so that two after spheres overlap at point A--can be driven as low as desired even if the locations of the after spheres are chosen randomly. Thus the molecular machine can choose an after state with little probability of error as long as the machine capacity is not exceeded. That this result is obtained from most random choices of the coding suggests that the evolution of good codes may be easy.

  

Figure 10: Correspondence Between Communication Theory and Molecular Machine Geometry.

The figure is the same as Figure 5 in [Shannon, 1949] except that the distances are given as $\sqrt{P_y}$, $\sqrt{N_y}$ and $\sqrt{P_y + N_y}$rather than $\sqrt{2 t WP}$, $\sqrt{2 t WN}$ and $\sqrt{2 t W(P + N)}$.

In Shannon's theory the capacity limit is approached by increasing t, while for the simple molecular machines described in this theory, d_space must increase. Molecular receivers, discussed in Appendix 23, could increase either t or d_space.