[Mike Dacre]

I have been thinking a lot recently about competitive and allosteric inhibition of enzymes. I am very new to the field, so it is possible that I am being stupid. I would appreciate the input of an actual biochemist who understands the math better than I do.

A little background about me: I am a career changing pre-medical student but I deeply hate the medical and pre-medical way of thinking about science. I have no desire to 'just memorize it and get in', I want to understand. I am taking a biochemistry class now, and some of the explanations of inhibition and allosteric regulation are unsatisfying to me, and my instructor is unable to resolve the issues for me. I am here in the hope of getting a good explanation from an actual biochemist. I am not afraid of calculus or molecular biology, so the more thorough the explanation the better. Thank you so much in advance.

I first started thinking about this while reading Voet (p.493) and thinking through the derivations of the Michaelis-Menten equations for competitive and uncompetitive inhibition. These equations are actually only conditional on a competitive inhibitor *blocking* substrate binding, it doesn't have to be by a blockage of the active site. Similarly, an uncompetitive inhibitor doesn't have to bind allosterically, it just has to bind the Michaelis complex in a way that inhibits the actual reaction without blocking substrate binding.

Despite the fact that textbooks and undergrad teachers everywhere claim that competitive inhibition is always by a substrate analog, there appears to me to be nothing in the math that requires this.

Following this line of thought and looking through the literature, I came across the concept of non-classical or allosteric competitive inhibition. A paper by Tippett and Neet in 1982 entitled 'An Allosteric Model for the Inhibition of Glucokinase by Long Chain Acyl Coenzyme A' describes an actual biological feedback loop that makes use of this allosteric competitive inhibition. Another paper (Postma and Catterall 1984) describes an allosteric competitive mechanism for a local anaesthetic, and there are many more similar papers.

There is also a good book from the NIH that discusses what competitive and uncompetitive inhibition actually are in the context of drug discovery: http://www.ncbi.nlm.nih.gov/books/NBK92001/

They make it quite clear that a competitive inhibitor does not need to be a substrate analog, nor does it need to bind the active site directly. It merely has to reversibly bind in such a way that is mutually exclusive with substrate binding.

Now the core of my confusion here has to do with allosteric regulation. None of my three textbooks (Stryer, Voet, Lehninger) really do a good job of explaining allosteric regulation, at least for me. My fundamental problem with the explanations is that it appears that there is a conflation of the properties of cooperative multi-subunit proteins, which follow n>1 Hill equations (and not the Michaelis-Menten equation (n=1)) with the role of allosteric regulation.

My simplistic understanding of an allosteric site is "a spot on the enzyme that isn't the active site, where some molecule binds either activating or inhibiting the enzyme". That activation/inhibition can come from a modification to the structure that prevents substrate binding or, more rarely, prevents a reaction from happening when substrate is bound. To me, it seems that inhibitory allosteric regulation is generally just non-classical competitive inhibition, or sometimes noncompetitive/uncompetitive inhibition. However, my three textbooks reject this saying that the enzyme kinetics of allosteric enzymes are 'fundamentally different' from examples of inhibition. That argument rings hollow to me, because the reason they cite for this 'fundamental difference' is cooperativity, and has nothing to do with the allosteric regulator itself.

From reading the UC Davis bio and chem wikis and learned that there are different types of allosteric inhibition, 'K systems' which affect K_0.5 but not V_max, and 'V systems' in which V_max is affected but K_M is not. To me this just sounds like competitive and noncompetitive inhibition from an allosteric site within a multi-subunit system. Is there a reason this is not the case?

This actually gets at something else that bothers me: ligand binding, enzyme kinetics, and allosteric regulation are all treated in my class as radically different processes with different equations that must be memorized, but the more I read and think about it, the more it feels like that just is not the case at all. These are the exact same processes with slight modifications in different systems. K_M is essentially the ratio of reactions that form a Michaelis complex to those that break it down, K_d expresses essentially the same thing but in a situation in which the ligand does not react. I know these are not directly comparable, but the similarity between the terms was just not emphasized in my course, and I feel that it was minimized in the textbook also. You already know my confusion over allosteric regulation vs reversible inhibition, and I feel that that falls into the same camp: extremely similar, if not identical processes, in different systems, used differently by the cell.

So my fundamental questions are: 'Is there a reason that allosteric regulation is not described using the same language as inhibition? Why are non-classical competitive inhibition, noncompetitive, and uncompetitive inhibition not just types of negative allosteric regulation (even if they are generally not a normal part of the system)? Are they really so fundamentally different that they cannot be described similarly? Is there a good, understandable, resource describing the kinetics of allosteric regulation that explains the differences with other forms of inhibition without invoking cooperativity?' These questions may be simple, but what I am really after is a deeper understanding of the regulation of proteins, particularly enzymes, but proteins in general.

Right now, I have a feeling that the fundamental difference comes down to the existence of tense and relaxed states in multi-subunit proteins. However, my understanding was that substrate binding generally induced the R-state and resulted in the expulsion of any allosteric factor that stabilized the T-state. Is that not the case? If it is the case, how is that not just mutually exclusive binding, aka competitive inhibition? Even if that is not the case, what about single subunit allosteric regulation (e.g. Gohara and Di Cera 2011)? In that paper if you define α as (1+r) then the effect of the allosteric regulation on the kinetics would be just αK_M, sounds familiar right?

I tried looking online for a satisfying answer already, but most discussion seems to be a little too vague, or is just people getting confused by the difference between uncompetitive inhibition and allosteric regulation. The literature has a tendency to focus on specific examples (understandably), and I sadly don't have the time to really work my way through all of the relevant papers on the subject to develop and understanding that way. In the absence of time or a good explanation from my textbooks/teachers/the internet, here I am.

Any help would be really appreciated. I don't need this for my exam or any homework, these questions are way above the level of either of those, so this isn't urgent. Any help that goes deeper than just 'competitive inhibition is defined as active site binding' (it really isn't) or 'allosteric regulation only works for multi-subunit proteins' (that doesn't answer the questions here) or "that's the way it is silly" (why?) would be appreciated. Journal references and/or deep explanations/mathematics are particularly welcome. Thanks so much everyone and apologies if I am just being really dumb and missing something obvious, that happens to me frequently.

References
-------------

Voet, Donald, and Judith G. Voet. 2010. Biochemistry, 4th Edition. 4th edition. Hoboken, NJ: Wiley.

Tippett, P. S., and K. E. Neet. 1982. “An Allosteric Model for the Inhibition of Glucokinase by Long Chain Acyl Coenzyme A.” Journal of Biological Chemistry 257 (21): 12846–52. http://www.jbc.org/content/257/21/12846.

Postma, S. W., and W. A. Catterall. 1984. “Inhibition of Binding of [3H]batrachotoxinin A 20-Alpha-Benzoate to Sodium Channels by Local Anesthetics.” Molecular Pharmacology 25 (2): 219–27. http://molpharm.aspetjournals.org/content/25/2/219.

Strelow, John, Walthere Dewe, Phillip W. Iversen, Harold B. Brooks, Jeffrey A. Radding, James McGee, and Jeffrey Weidner. 2004. “Mechanism of Action Assays for Enzymes.” In Assay Guidance Manual, edited by G. Sitta Sittampalam, Nathan P. Coussens, Henrike Nelson, Michelle Arkin, Douglas Auld, Chris Austin, Bruce Bejcek, et al. Bethesda (MD): Eli Lilly & Company and the National Center for Advancing Translational Sciences. http://www.ncbi.nlm.nih.gov/books/NBK92001/.

Nelson, David L., and Michael M. Cox. 2012. Lehninger Principles of Biochemistry. 6 edition. New York: W.H. Freeman.

Gohara, David W., and Enrico Di Cera. 2011. “Allostery in Trypsin-like Proteases Suggests New Therapeutic Strategies.” Trends in Biotechnology 29 (11): 577–85. doi:10.1016/j.tibtech.2011.06.001 http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3191250/.

[Babcock_Hall]

IMO it would be helpful to break down your post into smaller pieces, each one dealing with a specific question.  With respect to competitive inhibition, it only requires that the inhibitor bind mutually exclusively with respect to substrate, not that has to bind to the active site, let alone be a substrate analog.  However, there are probably more examples of competitive inhibitors that bind to the active site than inhibitors which do not.  K-systems usually show sigmoidal kinetics, whereas the only example of a V system I can think of offhand (acetyl CoA carboxylase) does not show sigmoidal kinetics.  In the case of allosteric regulation, the binding site for the regulatory molecule is often quite distant from the active site; however, I don't know any statistics which cover how frequently this is the case.

[Babcock_Hall]

I suggest reading Chapter 7 in Proteins: Concepts in Biochemistry, by Paulo Almeida.  The author discusses the case of oxygen and glycerate bisphosphate binding to hemoglobin from a perspective that you might find agreeable.  An introductory biochemistry textbook must make some simplifications that are not strictly true.  One typically teaches chemistry by successive approximations.

EDT
With respect to substrate analogs, and C, UC, and NC inhibition, I find it helpful to consider enzymes that have two substrates.  A substrate analog of substrate A will probably give competitive inhibition.  However, it might give either NC or UC inhibition with respect to substrate B, depending on the kinetic mechanism.

[Mike Dacre]

Thanks, and sorry for the very long question, I can be very verbose when I am thinking through a problem. I just purchased Proteins: Concepts in Biochemistry, by Paulo Almeida and it looks excellent, I intend to read it all. I particularly like the first part of the introduction:

Having taught biochemistry for about 20 years, I came to appreciate that students struggle with the sheer volume of material in introductory courses. They are asked to know an ever-growing amount of information (yet what a human being can reasonably learn and comprehend has not increased) at the expense of conceptual understanding. …
Biochemistry textbooks have become increasingly concentrated on biological aspects and on large amounts of information. Conversely, understanding chemical, physical, and mathematical aspects of biochemical topics have received less attention. Yet, already about 10 years ago the National Research Council (NRC) of the National Academies reached the conclusion that “life science majors must acquire a much stronger foundation in the physical sciences... than they now get,” and recommended increasing connections between biological and physical sciences as a means to bridge the communication gap that exists between researchers in different disciplines.

That is exactly how I feel about my pre-med education. I spend all my time memorizing pathways and I almost no time understanding how things work, I am expected just to memorize things, in the hope that I will get some understanding out of that, in the few months the memorized stuff lasts.

Right now I need to concentrate on memorizing metabolic pathways, but when my class is done I intend to spend a lot of time thinking about proteins and pathway regulation, because it is a fascinating topic, I really wish it was covered in more detail in my class. Thanks for the suggestions and I will try to keep my future questions brief.

[Mike Dacre]

One last thing, this is my current mental model of protein regulation, it is a little long. If anyone has time to read it and let me know if there is anything insane in there, that would be very helpful. Thanks again everyone :-)

Two ways of describing a protein binding site:
- ligand binding site/active site
- allosteric site

Regulation of protein activity (any alteration to binding or activity) can occur at either site. In non-enzymatic proteins, regulation is generally only on binding, but may also influence function (e.g. regulation of transporters). In enzymatic proteins, regulation is similar, with an proviso that changes to binding/protein activity will affect the rate of the catalyzed reaction, otherwise regulation is identical.

In all kinds of proteins, regulation can be either agonistic or antagonistic.

Regulation by the binding of a non-ligand/substrate molecule to the active/binding site is generally antagonistic (inhibitory), all forms of allosteric regulation can be either agonistic or antagonistic.

Either agonistic or antagonistic regulation can work on either ligand/substrate binding, protein/enzyme function (other than binding), or both.

Thinking now only of antagonistic regulation, there are four types:

- Competitive inhibition: regulator binds free protein in a way that precludes substrate binding
- Noncompetitive inhibition: regulator binds either free protein or protein-ligand/substrate complex in a way that blocks the additional functions (generally catalysis for an enzyme).
- Uncompetitive inhibition:  similar to noncompetitive inhibition except that inhibitor can bind only the protein-ligand/substrate complex and not the free protein
- Mixed inhibition: inhibitor binds both protein and protein-ligand/substrate complex in a way that simultaneously prevents ligand/substrate binding and also prevents additional functions (e.g. catalysis)

In my opinion all of those kinds of inhibition can occur at either the binding/active site or at an allosteric site, and the physical/kinetic effects would be essentially identical.

Furthermore, the above effect are independent of the cooperative effects of multi-subunit proteins. These cooperative effects result in a deviation from simple Michaelis-Menten kinetics and this deviation has nothing to do with allosteric or binding site regulation. It is however true that any regulatory effect will be more complex on multi-subunit proteins, as the underlying kinetics are more complex.