Assumptions

Assumption 1 Only some of the atoms in a molecular machine are involved in an operation. For example, if the flip of a tyrosine ring in bovine pancreatic trypsin inhibitor has no function [Karplus & McCammon, 1986] or effect on sphere sharpness or placement then d_space is effectively less than 3n - 6. In this paper d_space is taken to refer only to the number of spatial degrees of freedom involved in the operation. Even with the restriction of equation (2), d_space can still be large in a typical macromolecule, so (38) still applies.

Most protein dynamics are well modeled with just the locations of the nuclei, and quantum corrections are small at 300 $\mbox{K}$ [Levy et al., 1982,Ichiye & Karplus, 1987]. If quantum effects were used in a machine operation, d_space would be given by the number of independent parameters that are required to describe the system.

Two independent ``pins'' need not have the same importance to the organism. If we use information content as a measure of ``importance'', we can see that the ``importance'' of various bases in a binding site is strongly dependent on the position in the site [Schneider et al., 1986,Schneider, 1988,Schneider & Stephens, 1990]. Likewise, one pin in a lock could have more ``importance'' than other pins if it used more distinct levels than the others.

Assumption 2 The important parts of the molecular machine move independently. In the lock-and-key analogy, this assumption is that the pins of the lock move independently of one another. However, it is possible for one part of a molecular machine to affect the motions of its neighbors. In communications there are similar phenomena [Shannon, 1949]. Regions of a television picture are correlated to one another, and each frame is often similar to the next. Shannon pointed out that this simply reduces the number of independent parameters. So correlations between parts of the machine effectively reduce the dimensionality by confining the machine to surfaces in Y space. If the dimensionality is reduced, then C_y remains the upper bound, as can be seen from equations (37) and (38).

This assumption has a biological rational. It asserts that the components of a molecular machine can become independent through natural selection. For example, where it is important that two successive amino acids in the chain of a protein move independently to satisfy the protein's function, mutational insertions in the gene for the protein will confer a selective advantage. Eventually a flexible segment may evolve that allows the amino acids to move nearly independently.

The linear structure of binding sites on nucleic acids suggests that parts of the binding site recognizers could operate independently in the same sense that lock pins are independent. Three lines of evidence support this idea. First, it is possible to train a linear perceptron to identify ribosome binding sites and splice junctions [Stormo et al., 1982,Nakata et al., 1985,Brunak et al., 1990]. Second, it is possible to predict the amount of translational initiation using a linear model of the 12 bases preceding and including the first base of the initiation codon of ribosome binding sites [Childs et al., 1985,Stormo et al., 1986,Barrick et al., 1994]. There are similar data for the Cro, $\lambda$ and lac repressor binding sites [Takeda et al., 1989,Lehming et al., 1990]. Third, the contribution of individual amino acids to the total association free energy between proteins has been found to be additive in a number of cases [Horovitz, 1987,Horovitz & Rigbi, 1985]. The success of these approaches suggests that at least some parts of molecular machines exhibit independence and that further experimental work may allow us to map the locations of the ``pins''.

It is possible that a transformation of the descriptive variables is required to reveal independence. For example, if the transformation involved in harmonic analysis provides a good model for a particular molecular machine [Feynman et al., 1963,Colthup et al., 1975,Crawford, Jr. & Swanson, 1976,,Karplus & McCammon, 1979,Karplus, 1987,Ichiye & Karplus, 1987,Nishikawa & G $\bar{\mbox{o}}$ , 1987] then the modes are guaranteed to be independent, and the equipartition theorem [Waldram, 1985] guarantees that the energy is evenly distributed over all 3n-6 modes [Tidor et al., 1983]. A molecular machine need not use all of these modes.

The independence assumption has a curious consequence. Since its components are independent, the machine is modeled as an ideal gas in Y space and a machine operation is represented by the collapse of this gas. The entropy decrease is simply the log of the ratio of the volumes (equation (37)), as in classical thermodynamics [Castellan, 1971]. The decrease in entropy of the molecular machine is proportional to the information it gains.

Assumption 3 The energetics of molecular machine components (``pins'') are described by a Boltzmann distribution [Waldram, 1985,Colthup et al., 1975,Brillouin, 1962]. This is equivalent to assuming that each component is affected by band-limited white Gaussian noise [Nyquist, 1928,Rice, 1944,Rice, 1945,Shannon, 1949,Stremler, 1982,Petsko & Ringe, 1984] or Brownian motion [MacDonald, 1962] in which the velocity of a particle is the sum of many small impacts. Atomic fluctuations in proteins are well characterized by Gaussian distributions [Ichiye & Karplus, 1987].

Shannon considered the case of the channel capacity with an arbitrary type of noise [Shannon, 1949]. He pointed out that white Gaussian noise is the worst possible noise, and that other kinds of noise exist. As Shannon noted for communications systems, the ensemble states of molecular machines are not spherical when the noise is not white Gaussian. This is equivalent to changing the energy function of the ``pins''. For example, suppose that the energies were related to the maximum velocities x and y by

$\begin{displaymath}E_x \propto \vert x\vert^m \;\;\;\;\;\mbox{and}\;\;\;\;\; E_y \propto \vert y\vert^m \end{displaymath}$

(39)

instead of the form $E \propto x^2$ , as in equation (6). Then the total energy would be proportional to

|x|^m + |y|^m = |r|^m .

(40)

This may or may not be physically realizable, but we can use it to illustrate the possible properties of non-Gaussian noise. The case of m = 2 produces a circle, as in Fig. 3. This represents Gaussian noise. If m = 1 then the formula reduces to a line segment in the positive quadrant. This is reflected around the origin by the absolute value functions, to produce a ``diamond'' shape, as shown in Fig. 9. The figure also shows that there are a set of curves that lie between m = 1 and m = 2.

**Figure 9:** |x|^m + |y|^m = |r|^m .
$\vspace{11cm} \special{psfile=''fig/cisq.fig.ps'' hoffset=0 voffset=0 hscale=100 vscale=100 angle=0}$ The equation is plotted for m = 0.5 to m = 5 by increments of 0.1. Integer values of m are indicated by solid curves and other values by dotted curves.

If m > 2 then the curve bulges outward and the limit as $m \rightarrow \infty$ is a square! These shapes exceed the area of a circle with the same total energy. Now consider how these objects could be packed together. Circles could be packed into a hexagonal array. In contrast, the same hexagonal packing of the rounded squares would cause them to overlap, so circles produce a higher channel capacity. Since a molecular machine could obtain circles by evolving springs that move by simple harmonic motion, the m > 2 case could be avoided. This is why white Gaussian noise, where m = 2, is the worst possible noise. When m < 2 the area is less than that of a circle. At m = 1, the shape becomes a diamond and below this the shape is concave and has cusps. Since these spiky shapes can be packed more closely than circles, the capacity can be reduced in the absence of Gaussian noise. Similar effects occur in higher dimensions and with other force functions.

In general, if the effective ``entropy power'' of a noise N₁is less than the white Gaussian noise N_y ( $N_1 \le N_y$ ) then

$\begin{displaymath}\frac{P_y+N_y}{N_y} \le \frac{P_y+N_y}{N_1} \end{displaymath}$

(41)

so the machine capacity is bounded by

$\begin{displaymath}C_y \le d_{space}\log_{2} \left( \frac{P_y+N_y}{N_1} \right) \end{displaymath}$

(42)

and the upper bound exceeds the bound given by equation (38) [Shannon, 1949]. We can see this geometrically from the example given above. If the shape of the before state is spherical (i.e. the radius is $\sqrt{P_y + N_y}$ ), and the shape of the after state is spiky (i.e. the radius is effectively $\sqrt{N_1}$ ), then we obtain the upper bound of (42).

In this paper we have defined a classical physics benchmark against which we may examine real systems to see how well they do. Can a biological system use quantum effects to circumvent white Gaussian noise? By experimentally investigating the capacity of real molecular machines, it may be possible to answer this question.

Assumption 4 The before state is in equilibrium. The shape of the machine ensemble is spherical in the after state because the machine has reached equilibrium with its surroundings and the ``pins'' have a Boltzmann distribution (Assumption 3). In some cases the before state is also in equilibrium because the machine is a ``frustrated'' physical system [Shakhnovich & Gutin, 1989]. For example, on a time scale far shorter than it takes to find a binding site, a molecule of EcoRI should come to equilibrium with the surrounding solution. In contrast, if rhodopsin does not have a ``frustrated'' state, then one vibrational mode of rhodopsin might absorb more energy from a photon than the other modes, so that the ensemble would become an ellipsoid in Y space. However, of all possible ellipsoids, a sphere contains the largest possible volume given the constraint that the energy is constant. (For an ellipse, ${\left( \frac{x}{a} \right) }^2 + {\left( \frac{y}{b} \right)}^2 = r^2$ , the area, $\pi ab$ , is maximized when a = b.) So if the energies are unequally distributed in the before state, the volume will be smaller than that given by equations (36) and (26), M_y will be decreased (equation (37)), and hence the information gain, R, will be below C_y (equation (38)). Thus C_y remains the upper bound. We call this argument ``The Ellipsoidal Defense''.

It is advantageous for a molecular machine, such as rhodopsin or actomyosin, to operate as close to its capacity as possible, because then it would gain as much information as possible for a given energy dissipation. To operate near capacity, the machine must have, or equilibrate to, a spherical before state. In other words, the entropy of the before state will tend to be maximized by evolution, and the Ellipsoidal Defense is an argument that it is advantageous to the organism to allow the entropy of the before state to be maximized [Schneider et al., 1986,Schneider, 1988]. Indeed, there is evidence for ``complete thermal relaxation'' in the before state of rhodopsin [Hurley et al., 1977,Birge & Hubbard, 1980]. Complete thermal relaxation could easily be obtained by rhodopsin if it enters a ``frustrated'' state when excited by a photon. It is possible that this relaxation improves rhodopsin's capacity to detect light.

Assumption 6 The after spheres are perfectly packed and do not overlap. The after spheres could overlap. This effectively reduces the number of distinct after states M_y, and lowers the capacity according to equation (38). Thus C_y remains the upper bound.

Sphere overlaps represent transitions or isomerizations between semi-distinct states of the machine [Campbell et al., 1985,Ichiye & Karplus, 1987,Frauenfelder et al., 1988]. To see this, consider two after spheres that are so close together that they overlap. A point which is in the overlap region between the spheres could be considered to be part of either sphere. Now recall that each point in Y space represents a velocity configuration of the machine. A moment later the machine has moved, and this corresponds to a point somewhere else on the sphere. If the machine starts out on one of the spheres, and is in the overlap region next, it could easily end up on the other sphere. Since the other sphere represents a different after state of the machine, the machine would have two states but they would not be distinct because the machine would keep switching between them. The rate that the machine switches states depends on the volume of the overlap region relative to the size of the spheres.

These conformational substates may exist in either the before or the after machine states. If the before state is broken into several connected conformational substates, one can find a machine with a higher capacity by joining the substates, since this increases the volume of the before state. In contrast, if an after state is broken into several conformational substates, a better machine can be found by separating the substates, since this would increase the number of distinct after states and so increase the capacity of the machine.

As an example, suppose that an RNA polymerase inserts the four bases at W_s= 200 operations per second [Golomb & Chamberlin, 1974]. Since it performs $R = \log_2 4 = 2$ bits per operation, it operates at W_sR = 400bits per second, which we will take to be its capacity. (See Appendix 23 for a discussion of various forms of the capacity.) Now suppose that the temperature is raised, increasing the thermal noise and swelling the after spheres so that they overlap (equation (32)). Suppose that A and G become indistinguishable, that C becomes indistinguishable from U, but that the operating rate is not increased significantly by the temperature increase. Then the machine performs only 1 bit per operation at a rate of 200 bits per second. This shows how blurring the distinction between after states decreases the machine's rate of operation below the machine capacity.