158 J. D. BERNAL 



therefore are six objectively separate levels of organization, each one including 

 all those beneath it. 



This example brings out the main thesis of this paper, namely that: The 

 probability of formation of a highly complex structure from its elements is increased, 

 or the number of possible ways of doing it diminished, if the structure in question 

 can be broken down into a finite series of successively inclusive sub-structures. I 

 beUeve this theorem is capable of formal proof if it can be properly formulated, 

 but I leave this task to the logicians. Here I am concerned primarily to demon- 

 strate that such sub-structures can be formed from atoms related by the known 

 laws of physics and chemistry. Further, I want to show that each kind of sub- 

 structural unit corresponds to a definite and limited range of absolute sizes and 

 shapes and that at each level the laws of association of the units are qualita- 

 tively different. None of this, in principle, depends on biological analysis ; it 

 would hold for all combinations consisting mainly of the four elements hydrogen, 

 carbon, nitrogen and oxygen. As, however, only a few of these structures have 

 been produced synthetically, examples, particularly of the most compUcated 

 forms, will be sought in the field of biological structures. 



The size range, the shapes and the methods of mutual attachment of the imits 

 of every level depend on factors which are partly physical — that is in respect to 

 the nature of the mutual potentials or attractive or repulsive forces between the 

 particles — and partly geometrical, determined by their relative sizes and shapes. 

 It will be convenient to deal with this geometrical or formal element first, as it 

 is common to all sizes and natures of mutual potentials. Indeed formal analogies 

 to microbiological structures can even be found from human technical expe- 

 rience which in fact provides them with their vocabulary such as piling, twisting, 

 twining and pleating. A logical starting point is the quasi-spherical particle, for 

 others of different shape can always be constructed from such particles. All that 

 need be postulated of them is a more or less fixed radius and the capacity of 

 joining to one or more similar particles. The number that it can join on to may 

 be called its co-ordination number, analogous to the valency of atoms. The lower 

 limit of the co-ordination nvmiber is not necessarily fixed; that is a particle 

 can be hnked to fewer than it can accommodate but the upper number is hmited 

 by close-packing considerations to 12-14. 



In the simpler cases we are concerned with the Unking of similar particles or 

 at least, as in proteins and nucleic acids, of particles with similar Unkage systems 

 though with different side groups. More compUcated relations involving more 

 than one type of co-ordination can occur with heterogeneous particles and some 

 of these may be very important, such as those of the nucleoproteins, the Upo- 

 proteins and the mucoproteins, but the structure of these plainly depends on 

 that of their homogeneous components, which must be treated first. 



The co-ordination niunber of a particle with similar particles is not merely an 

 intrinsic property but also depends on external conditions and may be very 

 sensitive to them in cases where the difference in the free energy of association 

 of different kinds is smaU. 



How this can occur can be illustrated in the foUowing ideal case. In Fig. 3 

 (a and c) are shown two forms of the mutual potential surfaces of a pair of 



