## Friday Feature: Ceci n’est pas un film

June 18, 2010 § Leave a comment

It’s Friday — but this is not a movie. One must be fair to the non-imagers among us (theorists, biochemists and such). You are looking at a representation of the geometry of steady-state phosphoform distribution in a system consisting of two enzymes (a kinase and a phosphatase) and a substrate with two distinct phosphorylation sites, from the work described in Manrai, AK and Gunawardena, J. 2008. The geometry of multisite phosphorylation, *Biophys. J.* **95** 5533-5543. PMC2599844.

Why would you want to know about this? Post-translational regulation is a hugely important mechanism for changing the behavior or localization of proteins, and phosphorylation is possibly the most important form of post-translational regulation in eukaryotes. The number of phosphorylation sites on some of the proteins we study is staggering: the EGF receptor has 10, p53 has 16, and tau (the microtubule-associated protein in the fibrillary tangles in Alzheimer’s disease) has over 40. Since a protein with *n* phosphorylation sites has 2(*n*) possible ways of being phosphorylated, the presence of different phosphoforms adds enormously to the complexity of the mixtures of proteins found inside a cell.

How does a biological system interpret this complexity? p53 has an impressive variety of biological functions, but it’s hard to believe that the 2(16) different phosphoforms of p53 (>65,000) each have specific biological activities. It seems much more likely that cells use some kind of readout of the overall distribution of phosphorylations — possibly it’s the concentration of proteins with phosphorylations above a certain level that matters, or maybe it’s something more complicated than that. It’s hard to know without the tools to analyze phosphoform distributions.

Enter Manrai and Gunawardena. I will tread very lightly over the ground they cover: I can’t reproduce the mathematical reasoning in this format (especially since I’ve just discovered that I can’t even do superscripts in this particular WordPress style), but the entire Mathematica notebook with the proof is available if you want it. This is what they show:

— You can analytically solve the phosphoform distribution of a two-site system at steady state (without resorting to simulation) using tools from algebraic geometry including Gröbner basis methods. There is a helpful primer on Gröbner basis methods in the paper.

— When you perform this kind of analysis, you discover that the possible range of the phosphoform distributions has a different characteristic shape (geometry) depending on whether the enzymes (the kinase and phosphatase) in the system are processive, or distributive. [If an enzyme performs multiple operations — phosphorylations or dephosphorylations — on the substrate in the course of a single binding event, it’s processive. If it can only manage one operation before it drops off, it’s distributive.] This is the main thing that determines the shape of the curve along which the steady states can lie. The other factors that are important are the rate constants of the enzymes, and the initial conditions of substrate and enzyme concentrations.

The figure above shows an example of how the geometries of steady state distributions vary according to the nature of the enzyme pairs. The D/P, P/D, D/D and P/P traces in the figure are the curves on which the steady-state phosphoform distributions must lie if the kinase/phosphatase pair is distributive/processive, processive/distributive, both distributive or both processive. (The x, y, z axes represent different ratios of one phosphoform to another — see the paper if you want details — so each point is defined by three phosphoform ratios). Different enzyme parameters will vary the shape of the curve, and different initial conditions will place your steady state in different places along it. Interestingly, D/D systems always have steady states that lie on a plane, and either P/D or D/P systems can be thought of as deviating in specific ways from this plane.

What does this do for you as someone who wants to understand how biological systems interpret phosphorylation states? The first outcome, it seems to me, is really basic: this analysis motivates experiments that test what happens when whole enzyme/substrate systems are mixed (phosphatase, kinase and substrates). As the authors comment, these experiments have not often been done — which is understandable. Why would you want to obtain data that you know you can’t analyze?

Second — and this seems quite surprising, but it’s a theorem so it must be true — what this work says is that if you can distinguish the different phosphoforms in a system like this, you can start from a measurement of the steady state and work backwards to the behavior of the enzymes in the network (are the kinase and phosphatase pair processive or distributive or a mixture?) In other words you can infer the structure of the enzymatic network from its steady-state outputs.

It’s striking how little you need to know in order to make these inferences. You don’t need to know kinetic parameters for the enzymes: you just need to be able to manipulate kinase/phosphatase/substrate ratios and measure the phosphoform ratios that result. Admittedly, measuring phosphoforms is not trivial — yet — but mass spectrometry is rapidly getting to the point where it will be. And once you know the genre of kinase/phosphatase pair you’re dealing with, you know quite a lot about the phosphoform distributions that that enzyme pair is capable of producing. Assuming — as I do — that the phosphoform distribution is the key information read out by the downstream processes, this is a big step forward in understanding the outcome of the signaling events we’re studying. So far, the analysis has been done only for a two-site substrate, but there is nothing in the theory that suggests it cannot be expanded to more complex situations.

As a side note, let me highlight one issue that may look like a detail, but isn’t: Manrai and Gunawardena did not assume Michaelis-Menten kinetics, but derived their equations using mass action assumptions. Many models of kinase cascades do assume Michaelis-Menten kinetics, but this is only valid when:

• the reaction is irreversible with no product inhibition;

• there is a quasi steady state in the enzyme-substrate complex;

• the enzyme-substrate binding reaction has simple kinetics (not the case for multi-substrate reactions); and

• the level of enzyme doesn’t change over time

The last two of these clearly don’t apply to kinase cascades, and the others may not either. Beware: the difference between a mass-action-based model and a Michaelis-Menten based model can be dramatic. Here endeth the cautionary tale.

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