reaction in both directions (whereas the nature of a catalyst in itself means that it cannot catalyse just one direction, according to the principle of microscopic reversibility). We consider the case of an enzyme that catalyses the reaction in both directions: {E} + {S} <=>[k_{1}][k_{-1}] ES <=>[k_{2}][k_{-2}] {E} + {P} The steady-state, initial rate of the reaction is is positive if the reaction proceed in the forward direction () and negative otherwise. Equilibrium requires that , which occurs when . This shows that thermodynamics forces a relation between the values of the 4 rate constants. The values of the forward and backward maximal rates, obtained for , , and , , respectively, are and , respectively. Their ratio is not equal to the equilibrium constant, which implies that thermodynamics does not constrain the ratio of the maximal rates. This explains that enzymes can be much "better catalysts" (in terms of maximal rates) in one particular direction of the reaction. On can also derive the two Michaelis constants and . The Haldane equation is the relation . Therefore, thermodynamics constrains the ratio between the forward and backward values, not the ratio of values. Non-Michaelis–Menten kinetics Many different enzyme systems follow non Michaelis-Menten behavior. A select few examples include kinetics of self-catalytic enzymes, cooperative and allosteric enzymes, interfacial and intracellular enzymes, processive enzymes and so forth. Some enzymes produce a sigmoid v by [S] plot, which often indicates cooperative binding of substrate to the active site. This means that the binding of one substrate molecule affects the binding of subsequent substrate molecules. This behavior is most common in multimeric enzymes with several interacting active sites. Here, the mechanism of cooperation is similar to that of hemoglobin, with binding of substrate to one active site altering the affinity of the other active sites for substrate molecules. Positive cooperativity occurs when binding of the first substrate molecule increases the affinity of the other active sites for substrate. Negative cooperativity occurs when binding of the first substrate decreases the affinity of the enzyme for other substrate molecules. Allosteric enzymes include mammalian tyrosyl tRNA-synthetase, which shows negative cooperativity, and bacterial aspartate transcarbamoylase and phosphofructokinase, which show positive cooperativity. Cooperativity is surprisingly common and can help regulate the responses of enzymes to changes in the concentrations of their substrates. Positive cooperativity makes enzymes much more sensitive to [S] and their activities can show large changes over a