Unfortunately it is not a simple matter to translate Shannon's communications model into molecular biology. For example, his concepts of transmitter, channel, and signal do not obviously correspond to anything that EcoRI does or has. Yet, a correspondence exists between a receiver and this molecule since both choose particular states from among several possible alternatives, both dissipate energy to ensure that the correct choice is taken, both must undertake their task in the presence of thermal noise [Johnson, 1987], and therefore both fail at a finite rate (Appendix 21). By picking out a specific DNA sequence pattern, EcoRI acts like a tiny ``molecular machine'' capable of making decisions. Once the ``molecular machine'' concept has been defined, as best as is possible at present, we will begin to construct a general theory of how EcoRI and other molecular machines perform their precise actions. In doing this, we will derive a formula for the channel capacity of a molecular machine (or, more correctly, the machine capacity, equation (38)). The derivation has several distinct steps which parallel Shannon's logic [Shannon, 1949]. These steps are outlined below.
The lock-and-key analogy in biology draws a correspondence between the fitting of a key in a lock and the stereospecific fit between bio-molecules [Rastetter, 1983,Gilbert & Greenberg, 1984]. It accounts for many specific interactions. We will extend this analogy to include the moving ``pins'' in a lock, and then focus on each ``pin'' as if it were an independent particle undergoing Brownian motion.
To understand these motions, we consider simple harmonic motion of a particle, first in a vacuum and then in a thermal bath. The motion of many such particles serves as a model of how the important parts of a molecular machine (``pins'') move.
Just as any two numbers define a point on a plane and any three numbers define a single point in three-dimensional space, the set of numbers used to describe the configuration of the machine define a point in a high dimensional ``velocity configuration space''.
We then show that the set of all possible velocity configurations forms a sphere whose radius equals the square root of the thermal noise energy. Similar spheres appear in statistical mechanics as the Maxwell speed distribution of particles in a gas [Wannier, 1966,Castellan, 1971,Waldram, 1985].
When a molecular machine is primed, it gains energy and the sphere expands. When the molecular machine performs its specific action, it dissipates energy and the sphere shrinks while the sphere center moves to a new location. Because the location of the sphere describes the state of the molecular machine, the number of distinct actions that the machine could do depends on how many of the smaller spheres could fit into the bigger sphere without overlapping (Fig. 1). The logarithm of this number is the machine's capacity. Because the geometrical approach we take is the same as Shannon's approach [Shannon, 1949], his theorem about precision also applies to molecular machines. Hence, although molecular machines are tiny and immersed in a thermal maelstrom, they are capable of taking precise actions.
The enclosing large sphere represents a molecular machine having high energy, while each small sphere (gumball) represents the machine having low energy. There are many possible low energy conformations. The machine or channel capacity is the logarithm of the number of small spheres that can fit into the large sphere. |
The particular way that a molecular machine has evolved to pack the smaller spheres together corresponds to the way code words are arranged relative to one another in communications systems [Sloane, 1984,Cipra, 1990]. This suggests that we should be able to gain insight into how molecular machines work and how to design them by studying information and coding theory.