The model takes the form of an equation describing the rate of enzymatic reactions, by relating reaction rate v to [S], the concentration of a substrate S. Its formula is given by
v = Vmax [S]/Km + [S]
Here, represents the maximum rate achieved by the system, at maximum (saturating) substrate concentrations. The Michaelis constant Km is the substrate concentration at which the reaction rate is half of Vmax.
The Michaelis constant Km is the substrate concentration at which the reaction rate is at half-maximum,and is an inverse measure of the substrate's affinity for the enzyme—as a small Km indicates high affinity, meaning that the rate will approach Vmax more quickly. The value of Km is dependent on both the enzyme and the substrate, as well as conditions such as temperature and pH.
Therefore the relationship between Km and [S] is an inverse.
There is no Kmax, there is Vmax as shown in the above equation.
Km (the Michaelis constant), is a constant specific for a particular enzyme. As Km is a constant, it is not affected at all by increasing the substrate concentration. The relationship between Km and substrate concentration is that Km corresponds to the substrate concentration where the reaction rate of the enzyme-catalysed reaction is half of the maximum reaction rate Vmax. As the substrate concentration increases, the reaction rate will approach Vmax, but Km remains unchanged.
As others said, Km is a Michaelis-Menton constant and it won't change due to the level of substrate. However, it is an intrinsic characteristic of each enzyme and so, different enzymes and their isoenzymes have different metabolic behaviors and reaction rates.
'Km is a Michaelis-Menton constant and it won't change due to the level of substrate' - i.e it is specific for an enzyme with respect to the substrate in question. Just to add a small point to the earlier posts- substrate concentration or [S] should be >> Km so as to not become a limiting factor.