4. the dynamics of the epigenetic system 31 



Control Equations for Metabolites 



Now the concentration of metabolite, [A/,], was assumed to be control- 

 led by the concentration of protein, F,-. This means that y,is the rate-limiting 

 variable in the expression for the production of M,-. Restricting our attention 

 to the case where the metabolite M,- is the end-product of a reaction sequence 

 after which it feeds into a metabolic pool, we can write as the simplest con- 

 ceivable kinetic scheme for M,- an equation of the form (dropping now the 

 square-bracket notation) 



dMi ^ - , 



The term SiMi, s, a constant, imphes that M, is drawn off from the pool at a rate 

 dependent upon its own concentration; i.e. that this process is primarily 

 metabolite-controlled. The parameter /•,• represents a composite constant 

 which includes the rate constant for the enzyme 7,-, the concentration of its 

 substrate, and the Michaelis constant for the reaction whose product is M,-. 



It is at this point in the argument that we use the idea of relaxation times to 

 introduce a device which is frequently used in kinetic studies. Because the 

 variables 7,- and M,- belong to different systems as we have defined them in 

 Chapter 2, the first to the epigenetic and the second to the metabolic system, 

 their rates of change will be very different. Therefore, as has been argued 

 earlier, we can assume that the variable M, is always in a steady state relative 

 to significant changes in the variable 7,-, and so we can write 



^■ = r,7,-5,M, = (13) 



dt 



This allows one to solve for M, in terms of 7,-, and so to substitute /'.Tf/^, for 

 Mi wherever the latter variable occurs in the equations. This device of assum- 

 ing that certain reactions are in a steady state relative to others and thus 

 reducing the number of independent variables in the system is by no means a 

 new one, and has been used for many years in kinetic studies. Indeed the 

 original Michaelis-Menten formulation of enzyme kinetics depended upon 

 this procedure. Its vahdity, however, in this and other contexts has been 

 questioned, and the Briggs-Haldane treatment is one which does without the 

 assumption of an intermediate steady state in enzyme-catalysed reactions. The 

 purpose of discussing relaxation times in Chapter 2 was just to establish some 

 dynamic condition under which the use of steady state approximations is 

 justified, as well as to suggest a principle whereby different classes of dynamic 

 system can be operationally distinguished. 



If we consider feed-back inhibition in the kinetics of TW,-, so that this 

 product reduces the activity of 7,- in some manner, then we might have an 

 equation of the form 



dt C,+hiMi ' 



