Protein Structure and Information Content 119 



the probability density of a given microstate for molecular energy state E,^, 

 where the ranges of /,y and k can be essentially infinite. 



A transformation to a 'coarse-grain' scheme which seems worth consider- 

 ation is as follows. Each macrostate, M, (depicted by the leftmost column 

 of digits in Table III) designates only which bonds exist in the macromolecule, 

 e.g. sulfur atom no. 7 is hooked to carbon no. 179 and sulfur no. 11, C-563 

 to C-564 and N-201, etc. Mechanistically all microstates, w,;, contained 

 in a given macrostate, M,, are grouped together by ordering the digits (or 

 analogously ordering the axes in space). To complete the transformation 

 other bond properties, e.g. length and orientation (the other column of digits 

 in Table III), and their associated probabilities (the right hand portion of 

 Table III) are lumped into two gross categories to provide an intuitively manage- 

 able representation. This 'lumped fine structure' for each macrostate, Af,- 

 can be represented on an 'energy-deviation' {ED,) plane at the locus (in trans- 

 formed configuration space) corresponding to A/,: 'deviation' is a measure of 

 instabihty, i.e. the extent to which individual microstates, /77,^, deviate from 

 the configuration »7,^ corresponding to maximum stability for macrostate 

 Mj. An example of a method for constructing such values is: (a) find the set 

 of digits «?,s in the middle column of Table III which represents maximum 

 stability for macrostate M^ and (b) determine how many of the corresponding 

 digits of /;?,, and m^j differ. This number provides an excellent measure of 

 'deviation' because each microstate has a unique Z)-value and 'neighboring' 

 microstates have adjacent Z)-values. Assigning probabilities to pairs of 'energy' 

 and 'deviation' values completes the "fine" to 'coarse-grain' transformation. 

 This requires summing the probabilities, a,^;;,, of those microstates associated 

 with a particular D-value. The probability densities for E and D values can 

 be arranged into contours of equal probability to avoid further complications 

 of adding a third coordinate to the ED plane. These contours will possibly 

 be quite irregular in shape and may well be discontinuous, since the only 

 obvious restriction on their form is that they be non-intersecting. 



It should be noted that 'lumping' on to 'energy-entropy' planes would 

 have provided a simpler transformation than that to the 'energy-deviation' 

 planes. The microstates corresponding to a given 'deviation' can be equated 

 to an entropy value by the usual — S/^jlog/), procedure, where the /7/s are 

 the probabilities (properly normalized) associated with the microstates. Such 

 a scheme was considered, but was found to be intuitively less useful than the 

 ED transformation. 



The 'energy-deviation' scheme is of considerable interest when one con- 

 siders possible mechanisms of both protein inactivation and enzymatic activity. 

 Suppose, for instance, that the energy of a molecule in a native configuration 

 is slowly raised, e.g. by external heat: the point representing 'molecular state' 

 will be driven to new loci in multi-dimensional space. Undoubtedly a trajectory 

 is followed such that the locus resides, 'statistically', on the contour which 

 has the maximum probability permissible or consistent with its energy content 

 and macrostate at any instant. This means that the locus first progresses over 

 the EDj plane of the particular native configuration. A/,-. Eventually a locus 

 will be reached where the probability contour occupied is lower than the corre- 

 sponding contour on an adjacent ED plane. The molecular state will then 



