Hills Cooperativity Factor

One area where using SPR to study protein binding to DNA with multiple binding sites can be very useful is in estimation of Hill s cooperativity factor. According to Hill s theory [17,18], the simultaneous binding to DNA with n identical binding sites can be represented as  

After rearrangement and assuming that the concentration of free protein equals the concentration of injected protein C, Eq. 8 can written as  

After expressing the saturation fraction in the SPR terminology Y = Req/i ma the final form of Eq. 9 is  

For a successful analysis the response value at the equilibrium Req for each protein concentration must be known. This requires long protein-DNA contact times, which can be achieved by injecting protein solutions at a slow flow rate (mass transfer does not influence the Req value). The second important parameter for analysis is the response value at protein saturation Rmax- K can be difficult to reach, especially for negatively cooperative interaction. In this case the theoretical maximum binding value can be used  

MWdna = molecular weight of DNA MWprotein = molecular weight of protein 

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The equation says that the catalytic rate increases by the nth power of the substrate concentration when [Ao] is small in comparison to K. The Hill coefficient, n, is a measure of the sigmoidal character of the curve and, therefore, of the extent of the enzyme s cooperativity. For n = 1 (Equation 2.65) the reaction rate is transformed into the Michaelis-Menten equation, i. e. in which no cooperativity factor exists. 

Another factor that favours oscillations is the steepness of the function characterizing the repression of per transcription by nuclear PER. Such steepness depends on the degree of cooperativity, n, of the repression process, which is described in eqn (11.1a) by a Hill function. In agreement with such a destabilizing role of cooperativity, oscillations in fig. 11.7 were obtained for a value of n = 4 however, periodic behaviour can also occur in the model for n = 2 and for n = 1, but the domain in parameter space where sustained oscillations occur is smaller than for n = 4. 

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