4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 25 



reason for using this somewhat unorthodox measure of concentration will 

 become evident in Chapter 6. Our immediate goal is to establish a set of 

 extremely coarse functional relationships between the variables X,, Y„ and 

 Mi, which describe the essential dynamic features of the type of control system 

 which we have in mind. Then we will consider how far this rather nude model 

 can be dressed up to look biologically decent, and what are the inherent 

 limitations of this approach. 



Control Equations for Protein Synthesis 



Consider first an equation for the rate of synthesis of the /th species of 

 protein, when the controlling variable in this process is Xi, the concentration 

 of the corresponding species of mRNA . We will consider equations of the form 



^'' = UXi, 7,., M;)-g,(r,, M,.) (1) 



where /(X,-, 7;, M,) is a function describing the rate of synthesis of protein, 

 while gi{ Yi, M-) relates to the rate of its degradation. The simplest conceivable 

 functions which satisfy the control requirements of our model are given by 



MXi, Yi, M,) = a,. Xi, gii Yi, TV/,) = A- 



Then ^'=a,A',-i3, (2) 



Here ocjXi represents the rate of mRNA-controlIed protein synthesis, while 

 /S,- is the rate of degradation of the protein, assumed to be a constant. Since 

 protein synthesis is an almost completely irreversible process, no terms for the 

 reverse reaction on the template are included. We may note at this point that 

 control by catalysts in chemical systems actually depends upon irreversibility 

 in the reaction controlled, as has been emphasized by Elsasser (1958). The 

 reason for this is that at equilibrium a catalyst has no effect upon a chemical 

 process, and in order for the catalyst to act as a control " valve ", the reaction it 

 catalyses must be proceeding irreversibly. Thus biochemical control mechan- 

 isms can operate in open but not in closed systems. 



In equation (2) the constant a,- is a composite parameter containing a rate 

 constant for the template synthesis of the /th species of protein and concen- 

 tration terms for activated amino acids, the precursors of protein synthesis. 

 The simplification involved in using such a representation for what is clearly a 

 very complex biochemical process may seem to invalidate our analysis from 

 the start. The one feature of the real process which is incorporated in this 

 equation is control of protein synthesis by messenger RNA. Since it is the 

 dynamic consequences of this control which we seek to investigate, we will go 

 no further than equation (2) at present. The more general form of equation (1) 

 allows certain modifications to be included if they prove to be essential. 



