Michaelis Menten Kinetics- Identical Independent Binding Sites

Transcription

1 Michaelis Menten Kinetics- Identical Independent Binding Sites Dr. M. Vijayalakshmi School of Chemical and Biotechnology SASTRA University Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8

2 Table of Contents 1 INTRODUCTION SYMMETRY MODEL OF COOPERATIVITY SIGNIFICANCE OF THE HILL COEFFICIENT REFERENCE TEXT BOOKS... 8 Joint Initiative of IITs and IISc Funded by MHRD Page 2 of 8

3 1 Introduction Certain enzymes have multiple subunits each with identical binding sites. Yet there can be difference in the nature of binding sites, i.e., number of identical binding sites and the number of identical subunits of an enzyme. In such cases, the binding follows simple successive steps, where the first ligand binds to the enzyme, then second ligand binds, and so on. Consider a simple case where the enzyme E has n number of identical binding sites for the substrate A. The binding event is as follows: And it goes on till n sites are occupied. The corresponding expressions for macroscopic rate constants for the individual binding events can be given as: And for the n site, Imagine only there are 3 binding sites and thus the binding of A to site 1 will lead to EA 1, site 2 will be EA 2 and site 3 will be EA 3. The individual microscopic rate constants for the first binding event will now become: Joint Initiative of IITs and IISc Funded by MHRD Page 3 of 8

4 And hence we can arrive at If the binding sites are identical, And hence Similarly for second binding if we substitute the microscopic rate constants, where, Similarly make the expression for all the microscopic constants. Here, when all binding sites are identical, all the microscopic constants are equal to K and hence, Similarly for the third binding step, and applying the condition of identical binding sites, We can generalise the relationship between the macroscopic and microscopic rate constants for n binding sites as: Joint Initiative of IITs and IISc Funded by MHRD Page 4 of 8

5 The number of possible orientations given by Ω is also dependent on i A saturation function r is generally defined to make the binding equation simple. It is the quotient from the ratio of ligand bound to the enzyme to the total concentration of the macromolecule/enzyme. Substituting the following in r from the macroscopic rate expressions, Adair s Equation=> Using the binomial function, Joint Initiative of IITs and IISc Funded by MHRD Page 5 of 8

6 On application of binomial rule, the equation can be converted as: This on simplification gives the general binding equation. This equation resembles the Michaelis-Menten kinetics and can be graphically represented as follows. This expression when plotted as a direct or Scatchard plot, gives a direct measure of the number of binding sites (Fig1). Fig1: Direct plot and Scatchard plot giving information on the number of binding sites When the concentration of macromolecule is not known, the saturation function (Ȳ) is reduced by n and hence becomes, Ȳ at saturation reaches the value 1which gives the portion of ligand bound per binding site of the macromolecule. Joint Initiative of IITs and IISc Funded by MHRD Page 6 of 8

7 1.1 Symmetry Model of Cooperativity Jacques Monod and others presented a model to define the concept of cooperativity in case of allosteric enzymes. It is also referred as concerted model and it is based on few assumptions: Allosteric enzyme is an oligomer composed of n of identical protomers. Protomers occupy equal positions in the enzyme molecule and will possess atleast one symmetrical axis. Enzyme will exist in either of the two conformations, - tense(t) or relaxed(r) having diverse energy potential. No intermediate forms are present in the system. Symmetry is conserved in the reaction through transition from one form into the other. Tensed (T) state has low ligand affinity than the relaxed (R) state and has a low activity. This T state is favoured in the absence of ligand molecule. Fig 2. Concerted Model of cooperativity If there is no deviation from these assumptions, sigmoidal saturation curves are obtained. In a linear plot, significant deviation from straight line will be observed in cases where the co-operativity becomes apparent. So, Hill plot can be used to study such systems as it will clearly distinguish the two types of co operativity. The straight lines in the plot represents the binding of ligand to the two distinct states of enzyme T and R, while the distance between two lines represent the energy difference among the tensed and relaxed states. Joint Initiative of IITs and IISc Funded by MHRD Page 7 of 8

8 The maximum possible slope of the straight line is representative of the cooperative strength and is often referred as Hill Coefficient (n H ). Initially, the first ligand binds to the high affinity R form and the R state is now shifted from equilibrium. To restore it, the molecule from excess T form is transformed into R form. Then, when the second ligand comes, both forms of enzyme are present in equilibrium and there will be further free sites in R state that are disturbed from equilibrium. 1.2 Significance of the Hill coefficient Effectively the Hill coefficient ranges between 1 and n. When all the macromolecules are shifted to the relaxed form, the value of n H becomes 1. It describes the strength of cooperativity and if its value is in pair with the concentration of promoters, the cooperativity will be more evident. The Hill coefficient has a drawback that it does not directly state the number of subunits in the macromolecule. Thus, we understand that in the concerted model of Cooperativity, the equilibrium of 2 forms of enzymes depend on the allosteric centers. The next lecture will elaborate on the sequential model of Cooperativity. 2 Reference 2.1 Text Books 1. Bisswanger H, Enzyme Kinetics, Principles and Methods, WILEY-VCH, (2002). 2. J. D. Murray, Mathematical Biology, Springer-Verlag, (1989). 3. Berg JM, Tymoczko JL, Stryer L. Biochemistry, 5/e, W H Freeman, (2002). Joint Initiative of IITs and IISc Funded by MHRD Page 8 of 8