Karen M. Ong, John A. Blackford, Jr., Benjamin L. Kagan, S. Stoney Simons, Jr., and Carson C. Chow. A theoretical framework for gene induction and experimental comparisons PNAS 200911095; published ahead of print March 29, 2010, doi:10.1073/pnas.0911095107
This is an open access article so it can be downloaded directly from the PNAS website here.
This is a paper where group theory appears unexpectedly. The project grew out of a chance conversation between Stoney Simons and myself in 2004. I had arrived recently at the NIH and was invited to give a presentation at the NIDDK retreat. I spoke about how mathematics could be applied to obesity research and I’ll talk about the culmination of that effort in my invited plenary talk at the joint SIAM Life Sciences and Annual meeting this summer. Stoney gave a summary of his research on steroid-mediated gene induction. He showed some amazing data of the dose response curve of the amount of gene product induced for a given amount of steroid. In these experiments, different concentrations of steroid are added to cells in a dish and then after waiting awhile, the total amount of gene product in the form of luciferase is measured. Between steroid and gene product is a lot of messy and complicated biology starting with steroid binding to a steroid receptor, which translocates to the nucleus while interacting with many molecules on the way. The steroid-receptor complex then binds to DNA, which triggers a transcription cascade involving many other factors binding to DNA, which gives rise to mRNA, which is then translated into luciferase and then measured as photons. Amazingly, the dose response curve after all of this was fit almost perfectly ( > 95%) by a Michaelis-Menton or first order Hill function
where P is the amount of gene product, [S] is the concentration of steroid, Amax is the maximum possible amount of product and EC50 is the concentration of steroid giving half the maximum of product. Stoney also showed that Amax and EC50 could be moved in various directions by the addition of various cofactors. I remember thinking to myself during his talk that there must be a nice mathematical explanation for this. After my talk, Stoney came up to me and asked me if I thought I could model his data. We’ve been in sync like this ever since.
Stoney pointed me to an article written by Strickland and Loeb from 1981 and this article gave me most of the information I needed to proceed. The first thing I learned was how you get a first order Hill function for a dose response curve in the first place. The classical view of steroid-mediated gene induction was that steroid-receptor binding is the rate limiting step. So consider the reaction , where R is the receptor, S is the steroid and RS is the complex of the two. The goal is to derive an expression for the concentration of RS as a function of the concentration of S. If you assume the law of mass action then the probability of forming a product RS is just the probability of steroid S colliding with the receptor R and that is given by the product of their concentrations. Hence in equilibrium we have
where brackets indicate concentration and q is an association constant (inverse of the more commonly used dissociation constant). Now we need another equation to eliminate [R]. What we use is the fact that R is conserved so that
where is the total amount of receptor. Now, substitute (1) into (2), solve for [R] and substitute back into (2) and get
which is a first order Hill function with Amax = and EC50 = 1/q. If this reaction is the rate limiting step of gene induction then the gene product can be assumed to be proportional to [RS]. Hence, the classic view was that Amax and EC50 are dependent only on steroid-receptor binding. However, Stoney’s experiments showed that this view could not hold because cofactors acting downstream of steroid-receptor binding could move Amax and EC50.
Strickland and Loeb showed in their paper how Amax and EC50 of steroid-receptor binding could be moved by downstream reactions by exploiting an interesting property of the first order Hill equation, which is that the function is preserved under function composition. In other words if you substitute a first order Hill function into another first order Hill function, then the result is still a first order Hill function but with a new Amax and EC50. They argued that if there was a downstream reaction that had a dose response of the form
and then if we substitute (3) into (4), [F] would be a first order Hill function of [S] and Amax and EC50 will depend on downstream factors.
However, it was not clear to me why in a gene-induction process [X] would be proportional to [RS]. Strickland and Loeb were thinking in terms of reactions of the receptor-steroid complex before DNA binding. My picture was that RS would bind to DNA, which would then attract other transcription factors and you would get larger and larger complexes. Biochemically this could be expressed as a reaction cascade of the form , , , and so forth. However, if you solve for the dose response curve, for any number of reactions, you don’t get a first order Hill function (I didn’t have a full proof until much later). I then figured out that if downstream complexes had very small average concentrations or if cofactors only acted transiently but produced a lasting response then first order Hill form could be preserved for an arbitrary number of reactions. This then resulted in my first publication with Stoney, which can be downloaded here.
I then got busy thinking about other things and the project was kind of left hanging. However, in the summer of 2006 an enterprising undergraduate, Karen Ong, sent me an email. She was going to medical school in the fall but she took an advanced physics course as a senior and discovered that she loved math and physics. She was wondering if she could spend a summer in my lab and learn how to do biological modeling. I said sure and suggested that she re-examine Stoney’s problem. I was kind of dissatisfied with my first result because it didn’t pay any attention to the actual biology of the system. I thought Karen should try to construct a more biologically faithful model. As an example, she should try to model the action of the cofactor Ubc9 on gene induction. Stoney had found that Ubc9 moved EC50 strongly when the receptor concentration was high but not so much when it was low. So Karen set out that summer to consider how a chain of three reactions could reproduce the data. She was close to finding a solution near the end of the summer and decided to defer medical school for a year to stay in the lab and finish the project. She ended up staying for two years, eventually withdrawing from medical school and reapplying to MD-PhD programs. She is currently at NYU working towards an MD and a PhD in applied math at the Courant Institute.
Karen got two models working in the fall of 2006. One model involved transient reactions and the other consisted of complex building reactions. I had her learn Bayesian inference methods to fit and test the models. She eventually showed that the two models were equivalent if the complexes in the second model were small as expected by the theory.
However, I was still not satisfied because the models seemed too simple. Why did the other reactions known to be involved not matter. So I sat down one day, I think it was over Christmas holidays of 2007, but I’m not sure and wrote down the dose response formula in closed form for an arbitary cascade of n reactions. Two things allowed me to do this. The first was that I came up with better notation to represent complexes. The second and most important thing, which Strickland and Loeb did not mention in their paper so I don’t know if they knew this fact, is that the first order Hill function is in the family of linear fractional or Möbius transformations and forms a group under function composition. What this means is that function composition of first order Hill functions is equivalent to matrix multiplication of 2 by 2 matrices. So, calculating the formula of n function compositions is the same as calculating the product of n matrices and both could be done in closed form. I got quite excited by this result and emailed it to Karen right away. She then tinkered with it and came up with even better notation, which is close to what we use now. Over the next several months, we would send this LaTeX document back and forth adding to it each time. What started out as two pages eventually ballooned to over fifty. I also gave Karen a book on enzyme kinetics and she studied it religiously and used it to figure out how to incorporate inhibition into the reaction cascade among other things.
By the time I gave another retreat talk in April of 2008 we basically had our story. Preserving first order Hill functional form for the dose response curve puts a strong constraint on the system. We go into the biological implications in the paper. Mathematically, it implies that the system “telescopes” down to a small number of effective reactions depending on what you are interested in. So if you want to know the combined effects of three factors like steroid, receptor and Ubc9, then you only need to model a three reaction system. The other reactions are represented in the parameters of the effective reactions. So, you can always write down in rather simple form, the formula for the dose response curve for a finite number of factors. It took us another two years to get this paper published but that’s another story.
Addendum: Actually, at the April 2008 retreat, I told Stoney that dimerization of steroid-receptor complexes, which was believed to occur, would not yield a first order Hill dose response curve. Stoney was quite concerned about this since it was the central dogma and immediately started to design experiments to test it. He obtained and made several mutant receptors that could not dimerize and found that they gave similar results to the wild type receptors. This was probably the most crucial experiment to validate the theory in the paper.
Strickland S, Loeb JN (1981) Obligatory separation of hormone binding and biological response curves in systems dependent upon secondary mediators of hormone action. Proc Natl Acad Sci USA 78:1366–1370.
Loeb JN, Strickland S (1987) Hormone binding and coupled response relationships in systems dependent on the generation of secondary mediators. Mol Endocrinol 1:75–82.
Y. Kim, Y. Sun, C.C. Chow, Y.G. Pommier, and S.S.~Simons, `Effects of Acetylation, Polymerase Phosphorylation, and DNA Unwinding on Glucocorticoid Receptor Transactivation,’ Journal of Steroid Biochemistry and Molecular Biology 100, 3-17 (2006).