It may at first seem inconceivable that a purposeful sequence of operations, such as a computation, could be achieved in an apparatus whose direction of motion at any one time is nearly random. This style of operation is quite common, however, in the microscopic world of chemical reactions. There the trial-and-error action of Brownian motion, or random thermal motion, suffices to bring reactant molecules into contact, to orient and bend them into the specific conformation required for them to react, and to separate the product molecules after the reaction. All chemical reactions are in principle reversible: the same Brownian motion that accomplishes the forward reaction sometimes brings product molecules together and pushes them backward through the transition. In a state of equilibrium a backward reaction is just as likely to occur as a forward one.
In order to keep a reaction moving in the forward direction, we must supply reactant molecules and remove product molecules; in effect, we must provide a small driving force. When the driving force is very small, the reaction will take nearly as many backward steps as forward ones, but on the average it will move forward. In order to provide the driving force we must expend energy, but as in our ball-and-pipe realization of the Fredkin gate the total amount of energy can be as small as we wish; if we are willing to allow a long time for an operation, there is no minimum amount of energy that must be expended. The reason is that the total energy dissipated depends on the number of forward steps divided by the number of backward steps. (It is actually proportional to the logarithm of this ratio, but as the ratio increases or decreases so does its logarithm.) The slower the reaction moves forward, the smaller the ratio will be. (The analogy of the faster and slower swimmers is valid once again: it requires less total energy to go the same net number of reaction steps forward if the reaction moves slowly.)
We can see how a Brownian Turing machine might work by examining a Brownian tape-copying machine that already exists in nature: RNA polymerase, the enzyme that helps to construct RNA copies of the DNA constituting a gene. A single strand of DNA is much like the tape of a Turing machine. At each position along the strand there is one of four "bases": adenine, guanine, cytosine or thymine (abbreviated A, G, C and I). RNA is a similar chain like molecule whose four bases, adenine, guanine, cytosine and uracil (A. G. C and U) bind to "complementary" DNA bases.
The RNA polymerase catalyzes this pairing reaction. The DNA helix is normally surrounded by a solution containing a large number of nucleoside triphosphate molecules, each consisting of an RNA base linked to a sugar and a tail of three phosphate groups. The RNA-polymerase enzyme selects from the solution a single RNA base that is complementary to the base about to be copied on the DNA strand. It then attaches the new base to the end of the growing RNA strand and releases two of the phosphates into the surrounding solution as a free pyrophosphate ion. Then the enzyme shifts forward one notch along the strand of DNA in preparation for attaching the next RNA base. The result is a strand of RNA that is complementary to the template strand of DNA. Without RNA polymerase this set of reactions would occur very slowly, and there would be little guarantee that the RNA and DNA molecules would be complementary.
The reactions are reversible: sometimes the enzyme takes up a free pyrophosphate ion, combines it with the last base on the RNA strand and re-leases the resulting nucleoside triphosphate molecule into the surrounding solution, meanwhile backing up one notch along the DNA strand. At equilibrium, forward and backward steps would occur with equal frequency; normally other metabolic processes drive the reaction forward by removing pyrophosphate and supplying the four kinds of nucleoside triphosphate.