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Definition of Molecular Machines

In each example given in the previous section, a specific macromolecule is primed from a low energy level--or ground state--into a high energy state. This is followed by a specific action that dissipates the energy and performs a function that is evolutionarily advantageous to the organism that synthesized the macromolecule. There are many other examples of molecular machines that follow this pattern [Porter et al., 1983,Watson et al., 1987]. In general we will not be interested in the priming step, but rather with a precise measure of the specific action taken in exchange for the lost energy. The measure we will use is the number of distinct states which the machine can choose between. If the machine can select from two states, we say that it gains 1 bit of information per operation. Likewise, the selection of one state from amongst 8 corresponds to $\log_2 8 = 3$ bits per operation [Pierce, 1980].

1.

A molecular machine is a single macromolecule or macromolecular complex. In this paper we discuss the microscopic nature of individual molecules, not the macroscopic effects of large numbers of molecules. A molecular machine is not a macroscopic chemical reaction [McClare, 1971]. This does not deny that we can model a solution containing many molecules of EcoRI and DNA (without magnesium) by stating that the ratio of specifically bound to non-specifically bound molecules is constant once the reaction has reached equilibrium. This binding constant reflects the energetics of the individual reactions ( $\Delta G^{\circ}$), but it does not reveal the binding mechanism because that is independent of concentration. A single EcoRI molecule will cut a single DNA molecule irrespective of the number of other DNA and EcoRI molecules in the solution.

Suppose, for example, that we allow a macroscopic solution of DNA and EcoRI (without magnesium) to come to equilibrium at 37 ${}^\circ \mbox{C}$. Since individual molecules continue to bind and disassociate under these conditions, machine operations take place even after macroscopic equilibrium has been reached [Conrad, 1985]. Thus, the operation of a single molecular machine cannot be treated as a macroscopic chemical reaction since that ``stops'' when equilibrium is reached. For this reason, the molecular machine model does not (and should not) refer to concentrations.

As McClare [McClare, 1971] pointed out, each molecular machine acts locally as an individual. Likewise Arrhenius et al. [Arrhenius et al., 1986] distinguish functions at the molecular level from bulk material effects.

It is also worth noting that EcoRI alone is not a molecular machine. Only the combination of EcoRI and DNA is a molecular machine. Likewise, only the combination of a car and a road (or other suitable surface) can do useful work.

2.

A molecular machine performs a specific function for a living system. That is, if the machine did not exist, the organism would be at a competitive disadvantage relative to an organism that had the machine. Thus, a molecular machine must be important for the evolutionary survival of an organism or it will be lost by atrophy. Shannon pointed out that information theory is unable to deal with the meaning or value of a communication [Shannon, 1948,Shannon & Weaver, 1949]. In biology, however, we work with the closely related concepts of function and usefulness, factors which are ultimately defined by natural selection. This part of the definition is important for accounting for the precision of molecular machines. Without a requirement for function, precision--or any other non-deleterious property--does not matter, just as nobody cares whether or not a car on a junk heap works. With a requirement for function, the very survival of the organism is it stake. In practical terms, the requirement for precise function dictates that the states of the molecular machine should be distinct and hence that the spheres represented by gumballs in Fig. 1 should avoid overlap.

This definition encompasses machines that operate outside cells, such as digestion enzymes, and machines created entirely by humans [Drexler, 1981,Drexler, 1986].

(Even a Rube Goldberg² molecular machine's function would be to amuse, to educate, or to attempt to evade this definition.) Unlike simple chemicals like water, molecular machines are usually encoded by a genetic material and have the potential to evolve by natural selection.

3.

A molecular machine is usually primed by an energy source. These include not only photons and ATP, but also thermal motions--as in the case of EcoRI separating from a binding site. (DNA heat-denaturation is an artificial method that only appears in the laboratory. Natural priming mechanisms usually do not use this macroscopic heating, although they frequently use the ``microscopic heating'' provided by thermal fluctuations.) Priming places the machine in an activated before state where it is ready to do work. The before state corresponds to the large sphere that encases the gumballs in Fig. 1.

The act of priming is usually, but not always, required for a molecular machine to operate. For example, just after a new molecule of EcoRI has been synthesized, it is ready to operate even though it never was in a low energy state before.

4.

A molecular machine dissipates energy as it does something specific. This phase of the machine's cycle is called its operation. Once the operation is completed, the machine is in an after state, which is represented by a single gumball in Fig. 1. Since the machine is always subject to thermal noise, an after state consists of the set of all possible motions that a single molecular machine could have at low energy. We will call this set an ensemble. Likewise the before state consists of the set of all possible motions that a single molecular machine could have at high energy, and this also forms an ensemble.

5.

A molecular machine ``gains'' information by selecting between two or more after states. For example, EcoRI chooses one pattern out of 4⁶ = 4096 possible hexa-nucleotides, so it gains $\log_2 4096 = 12$ bits of information during its operation. Measurements of the amounts of information gained by genetic recognizers have been described in previous papers [Schneider et al., 1986,Schneider, 1988,Schneider & Stormo, 1989].

6.

Molecular machines are isothermal engines, not heat engines [Jaynes, 1988]. They are obliged to operate at a single temperature because they do not have any way to insulate themselves from the huge heat bath that they are embedded in. However, they can use a priming energy to change their conformation to a more flexible one. This is essentially a controlled form of denaturation. After priming, any excess energy is quickly dissipated, leaving the molecule trapped in a flexible before state at the ambient temperature. In this state the machine is like a ``frustrated'' physical system [Shakhnovich & Gutin, 1989] randomly searching through various conformations to find the correct one for the operation. When this is found, the formerly inaccessible (i.e. potential) energy is quickly dissipated leaving the molecule once again at ambient temperature. This model allows for the evolution of a molecular machine from primitive beginnings because the energy is captured by a denaturation, which is simple and easy to achieve. The model does not require any form of molecular insulation or special vibrational modes which would be difficult if not impossible to evolve.

This paper shows that the number of parts of a machine, the energy dissipated per operation and the thermal energy in the machine determine the largest amount of information a molecular machine can gain (equation (38)). This ``channel capacity of a molecular machine'' (or, more accurately, ``machine capacity'') is measured in bits per operation, where one bit is the amount of information necessary to choose cleanly between two distinct machine states. This paper demonstrates that although the machine capacity is sharply limited by the amounts of dissipation and the thermal noise, the accuracy of the machine is not.