The key question is whether your model is overdetermined (too many parameters). You can determine whether adding parameters improves the fit in a significant way by carrying out an F-test, which compares fits with N vs N+1 parameters and establishes whether the additional parameter results in a statistically significant improvement in the fit and parameter estimates. See Beck, J. V., and K. J. Arnold. 1977. Parameter Estimation in Engineering and Science. Wiley, New York.

The key question is whether your model is overdetermined (too many parameters). You can determine whether adding parameters improves the fit in a significant way by carrying out an F-test, which compares fits with N vs N+1 parameters and establishes whether the additional parameter results in a statistically significant improvement in the fit and parameter estimates. See Beck, J. V., and K. J. Arnold. 1977. Parameter Estimation in Engineering and Science. Wiley, New York.

Cortes et al (2001 Biochem J 357 263-268) commented that the binding of inhibitors to enzymes is "often cooperative" and used an equation in which both [I] and Ki are raised to the power h. We (Wei et al 2005 JBC 280, 32879-32882) observed cooperativity in the inhibition of TNF-alpha converting enzyme by TIMP-3 and essentially followed their treatment.

Thank you both for replying so quickly! This information will be really help. I will have a look at the papers mentioned.

Thanks

I think there are two elements to consider here. The first is the inhibitor mode of inhibition. Your equation assumes exclusively competitive inhibition. It is possible that the inhibitors are showing partial non-competitive binding (e.g. bind to the substrate-bound form), in which case it is appropriate to incorporate a variable for the cooperativity (usually termed alpha). However, this requires including a [S]*KmObs term in the denominator of your equation, and alpha is a factor of Ki only, not the [I] term. For a detailed treatment of this, see Chapter 8 in the book by Copeland, "Enzymes: A Practical Introduction to Structure, Mechanism, and Data Analysis", 2nd Ed, Wiley, NY, 2000.

Second, iti s possible that the stoichiometry of binding is not 1:1 and the observed cooperativity is due to multiple compound binding events. In this case, the equation is correct as written, but you should consider wheterh it is plausible for multiple binding events for a competitive inhibitors (e.g. the substrate binding pocket allows more than one inhibitor binding site). For large multimeric protein systems cooperative binding is often seen and can result in Hill slope effects.

However, in my experience, if the system is relatively simple cooperative effects are generally a sign of promisicuous, non-specific aggregation inhibition and not robust reversible inhibitors. For examples of this type of inhibitors search for papers by Brian Shoichet, UCSF and the term "promiscuous". Note that these inhibitors often show time dependence.

In the equation you are writing there is an implicit introduction of infinite cooperative of binding of the inhibitor. In other words, one the first site is occupied, the affinity of the second, third ... sites will be so high that these will be occupied. Basically you have EI_n, E_free or ES. Theoretically, therefore there is no one Ki, but as many as sites. If you apply the exponent to the Ki your KI is an apparent constant representing I50= (KI_low*KI_med...KI_high)^(1/h). Where KI_low represents the affinity of binding to the first site and KI_high represents the affinity of binding to the last site. In other words, once you introduce the Hill's infinite cooperativity approach to the binding of the inhibitor KI becomes an apparent constant. Therefore, you must be careful when trying to interpret your results.

The model you are implying is (total inhibition & initial velocity conditions):

E + S E.S --k_cat--> E + products

+

n I

^

KI' |

v

E.I_n

Under rapid equilibrium and assuming no reverse reaction (absence of product) it gives an equation:

eq.1) v_0= k_cat [E_t] [S] /(K_S (1+ [I]^n/KI') + [S] )

But this is simplification of

E + S E.S --k_cat--> E + products

+

I

^

KI1 |

v

E.I

+

I

^

KI2 |

v

E.I_2

+

I

...

^

KIn |

v

E.I_n

where

KI' = [E][I]^n/[E.I_n] = ([E][I]/[E.I]) ([E.I][I]/[E.I_2]) ... ([E.I_n-1][I]/[E.I_n]) = (KI_1) (KI_2)...(KI_n)

Since KI' is the product of many constants you can obtain an average measure of the affinities by taking

I50=KI'^(1/h)= ((KI_1) (KI_2)...(KI_n))^(1/h)

This value (I50) represents the concentration of inhibitor resulting in a duplication of the apparent K_S value, as it can be deduced from eq. 1.

Best wishes

Rogelio

Allosteric inhibition is perfectly compatible, depends on the type of enzyme also to consider if it applies or not. To assess if the model is overdetermined about the number of parameters, you may apply Akaike's criterion that considers both the RSS and the number of parameters of your model.

I have tried myself complex models for Acetylcholinesterase, self-inhibition by substrate, and modulation by polyamines (natural aliphatic amines present in all cells). The references are:

Rosenbaum E.A., D'Angelo A.M., Bergoc R.M. and Venturino A. (1999): "Modelling acetylcholinesterase kinetics: the identifiability problem in parameter estimation". J. Biol. Sys. 7: 95-111. ISSN 0218-3390.

Venturino A., Bergoc R.M., Pechen de D’Angelo A.M. and Rosenbaum E.A. (2002): “Kinetic models on acetylcholinesterase modulation by self-substrate and polyamines. Estimation of interaction parameters and rate constants for free and acetylated states of the enzyme”. J. Biol. Sys. 10: 127-147. ISSN 0218-3390.

Hill Equation in simplicity is just a Michaelis-Menten (MM) Equation where [S] and Km are raised to power n. If n is greater than 1 it is positive cooperativity and if n is less than 1 its negative cooperativity. If n = 1, it reduces to MM Equation.

I use ENZYME KINETICS MODULE EK1 linked to Sigma-Plot. It model all types of enzyme kinetics data including Hill, various types of inhibition, activation etc and ranks them.

If you have a suitable method available, you might want to try to measure the binding stoichiometry and affinity of the inhibitors directly, such as by equilibrium dialysis, isothermal titration calorimetry, or spectroscopy. That should help you decide whether there is more than one binding site for the inhibitor, which is obviously a necessity for cooperativity.

In the kinetic version of the Hill equation, Ys = V/Vmax = (S)^h/[KM + (S)^h]

this is (S) and not KM which is raised to the power of h.

[also, I would prefer to write Ks (the substrate dissociation constant) rather

than KM because a rapid-equilibrium hypothesis must be made to justify the equation from binding cooperativity of S]

One limitation of this equation is that it cannot be used without the knowledge of Vmax (which requires measurements at saturating conc of activator).

As for your inhibitor I, if we exclude slow-binding and assume positive binding cooperativity with another Hill coef h', we can write the saturation function as follows :

Yi = Sum(Ei)/Sum(sites) = (I)^h'/(Ki + (I)^h')

then write the left term of this eq more explicitely depending on the

inhibitory mechanism (competitive, purely non competitive etc).

But again here, the Hill coef h' will apply to (I), not to Ki

Finally, applying Hill coefs to rate constants has no physical meaning. This would be pure mathematical manipulation which might turn out to be convenient in terms of non-linear adjustment, but the coef would actually not reflect how strong is the cooperativity.

One analytical model in which Ks or Ki are raised to powers is the sequential-interaction model of Pauling and Adair, but here the coefs are always integers (1, 2, 3 ... n) which is not the case of Hill coefs, and the resulting functions are rational polynomials which are more complex than your expressions.

I agree with Deborah Leckband concerning non-linear adjustment and curve fits which look good but are mathematically unacceptable: if you use the Marquardt algorithm (most commercial softwares use it) , an F-test will enable you to check whether a parameter is necessary or not, and an overdetermined system will not converge on a stable solution. Although you might get a good fit after a fixed number of iterations is reached, the program will tell you that the result is statistically unacceptable.

I also agree with Adam Shapiro, the most important thing is to collect binding data that would support some kind of model. If you do not have radiolabeled ligands, the use of the chromatographic method of Hummel & Dreyer and/or that of isothermal titration calorimetry might be good alternatives.

The use of the competitive inhibition, noncompetitive and mixed competitive equations as a whole is flawed so applying the hill coefficient to the competivite inhibition equation just results in an even more flawed way of looking at enzyme inhibition.

The 1+[I]/Ki term that is used in the noncompetitive and competitive inhibition equations is not the same term in these equations. This is easily seen by the fact the competitive equation only relates to change in substrate affinity and is described by the inhibition term directly multiplying the substrate affinity term (Km). The same term when used to describe inhibition that affects the maximum velocity divides into the Vmax. Therefore inhibition terms are inversely affecting the terms of the MM equation. A rearrangement of the noncompetitive term results in a term that looks exactly like the MM equation

Vmax X (1-([I]/[I]+Ki))

where the reduction of enzymatic turnover rate is directly linked to the % of the enzyme population bound by the inhibitor. The same rearrangement of the competitive inhibition equation produces nonsense

Km/(1-([I]/[I]+Ki))

By realizing that the affect on substrate affinity must be directly related to the % of the enzyme population bound by the inhibitor a single equation can be used to describe inhibition that affects the Vmax, Km or both at the same time

v = Vmax - dVmax ([I]/[I]+Ki)) X [S]/([S]+(Km - dKm ([I]/[I]+Ki)))

The change in Vmax or Km can not be assumed to be absolute so a delta term must be included, however if experimentally there is no affect on the Km and the inhibition of the Vmax was found to be 100% this equation could reduce to the traditional noncompetitive inhibition equation. The presence of delta terms also means this equation can be used to describe activation as well.

While this doesn't help you determine whether applying the hill equation to the competitive inhibition equation is right or not it hopefully offers you an alternative way of modeling your kinetics which doesn't try to graft flawed mechanistic models to changes in MM kinetics which should be determined experimentally.

Hi everyone, thanks so much for all your helpful answers. I apologise for the late reply – I have been preparing for and attending a conference over the last week.

Stewart, to begin fitting curves to my inhibitory assay data, I used a mixed inhibition equation (http://www.graphpad.com/guides/prism/6/curve-fitting/index.htm?reg_mixered_model.htm) before then individually fitting equations for competitive, noncompetitive and uncompetitive inhibition modes. By comparing the R square values and the agreement between the mixed model fit and the specific model fits, I diagnosed the inhibition mode. There doesn’t seem to be any significant mixed inhibition in the case of the first inhibitor. I have just obtained crystallographic data that shows only one molecule bound in the active site of the enzyme. However, the inhibitor does contain two aldehyde groups, so there is a possibility that there is a covalent binding event that needs to be included in the model. Thanks for the suggestions on the Copeland book and the Shoichet papers. The Copeland book has been my main reference and I have come across some of the Shoichet papers before, but I think they will come in handy in the future.

Thanks, Rogelio. In this case, case it would seem appropriate to apply the hill coefficient to the Ki, and maybe this is the approach taken by other researchers when they have published such equations. However, you have rightly pointed out that the Ki, would now be an apparent Ki, so I should take care when reporting a final value.

Thanks for the references, Andrés. My enzyme also shows substrate/product inhibition, so I will have to model that, too.

Thanks for your insight, Jean. I was wondering if, in the case of equations, such as the slow-binding inhibition equations, where an apparent Ki is used, would it be appropriate to apply a hill coefficient?

Either way, I think my next move is to look into the F-test and Aikekes criteria to assess the feasibility of my equations. I can get access to an isothermal titration calorimeter, so I can hopefully use that to clear things up.

Some more info on the enzyme I’m studying may make things clearer to everyone: the enzyme catalyses a two substrate reaction by a Bi Bi compulsory ordered mechanism. Cooperativity has been observed when varying the concentration of one of the substrates for this enzyme. The enzyme is a homodimer, so cooperative interactions can exist between the active sites on each subunit. The picture gets even more complicated: size-exclusion chromatography experiments found that, on the addition of the second substrate to the enzyme, there was a shift in the elution volume from one corresponding to a dimer, to one corresponding to a monomer, suggesting that there is some dissociation. The [substrate] v. v plot produced a double sigmoidal curve. In case you are unfamiliar with this phenomenon, the following paper shows some examples of proteins that show this behaviour: http://www.sciencedirect.com/science/article/pii/S0003986111003948 .

The standard inhibitory equations I am using serve as good starting points to identify the MOA of individual inhibitors. However, I need to find the most appropriate equations, so that I can compare different inhibitors (I’m hoping to run a medium-throughput screen in the next month or two).

Could it be the case that an inhibitor could show signs of cooperativity due to its disruption of the cooperative effects of the substrate? The double-sigmoidal curve shown in the attached file was modelled by applying the hill equation to the first and second portions of the curve. I want to try and fit equations from the Monod, Wyman & Changeux or dissociating oligomer theories of cooperativity. If anyone has any references for such equations, particularly for dissociating oligomers, I would be very grateful.

Thanks Ryan, for your suggestions. The alternative equation you’ve presented could actually work very well with my data.

Thanks.

Eyram