Modelling success stories (2) Monod-Wyman-Changeux 1965

For the second model of this series, I will break my own rule limiting the topic to “systems-biology-like” models, i.e. models that are simulated with a computer to predict the behaviours of systems. However, a fair number of MWC models resulted in the instantiation of kinetics simulations, so I do not feel too bad about this breach. The reason to include the MWC model here is mainly because I think the work is one of the early examples where a model shed light on biochemical processes and led to a mechanism, rather than merely fit the results.

The model itself is described in a highly cited paper (5776 times according to Google Scholar on March 14th 2013):

Monod J, Wyman J, Changeux JP. On the nature of allosteric transitions: A plausible model. J Mol Biol 1965, 12: 88-118. PDF2

Contrarily to the Hodgkin-Huxley model, described earlier in this post, the main body of the work is located in a single page, the fourth of the paper. The rest of the paper is certainly interesting, and several thesis (or even careers) have been devoted to the analysis of a formula or a figure found in the other pages (several papers were even focused on the various footnotes, the discussions still going on after 50 years). However, the magic is entirely contained in this fourth page.

Cooperativity of binding had been known for a long time, ever since the work of Christian Bohr (the father of Niels Bohr, the quantum physicist) on binding of oxygen to hemoglobin. For an historical account see this article, to be published in PLoS computational biology and then on Wikipedia. Around the year 1960, it was discovered that enzymes also exhibited this kind of ultrasensitive behaviour. In particular the multimeric “allosteric” enzymes, where regulators bind to sites sterically distinct from the substrate, displayed positive cooperativity for the regulation. At that time, the explanations of the cooperativity relied on the Adair-Klotz paradigm, that postulated a progressive increase of affinity as the ligand bound more sites, or the Pauling one, based on only one microscopic affinity and an energy component coming from subunit interactions. In both cases, the mechanisms are inductionist, the ligand “instructing” the protein to change its binding site affinities or its inter-subunit interactions. In addition, the state function (the fraction of active proteins) and the binding function (the fraction of protein bound to the ligand) were identical (more exactly there was not even the notion that two different functions existed), something that was shown to be wrong for the enzymes.

The model developed by Monod and Changeux (Jeffrey Wyman always referred to the paper as “the Monod and Changeux paper”) relied on brutally simple and physically based assumptions:

  1. thermodynamic equilibrium: the proteins which activities are regulated by the binding of ligands exist in different interconvertible conformations, in thermodynamic equilibrium, even in the absence of ligand. This assumption is opposed to the induced-fit mechanism whereby the protein always exists in a conformation in the absence of ligand, and is always in the other conformation when bound to the ligand.
  2. different affinities for the two states: the two conformations display different affinities for the ligand. Consequently, the ligand will shift the equilibrium towards the state with the highest affinity (that is the lowest free energy). This is a selectionist mechanism rather than instructionist. The binding of a ligand no longer provoke the switch of conformation. Proteins flicker, with or without the ligand bound. However, the time spent in any given conformation depends on the presence of ligand (or the probability to be in a given conformation).
  3. all monomers of a multimer are in the same conformation: this assumption was, and still is, the most controversial. It is opposed to the notion of sequential transitions, whereby the monomers switch conformation progressively, as the ligands binds to them.

The rest followed from simple thermodynamics, explained by the two figures below.

MWC reaction scheme

Reaction scheme showing the binding of ligands to an allosteric dimer. c=KR/KT.

MWC energy diagram

Energy diagram showing the stabilisation effect of successive binding events.


The MWC model has been successfully used to explain the behaviour of many proteins, such as hemoglobin or allosteric enzymes, as mentioned above, but also neurotransmitter receptors, transcription factors, intracellular signalling mediators or scaffolding proteins. For an example of how MWC thinking help to understand signalling cascades, see our work on calcium signalling in synaptic function (Stefan et al. PNAS 2008, 105: 10768-10773; Stefan et al.  PLoS ONE 2012, 7(1): e29406; Li et al. PLoS ONE (2012), 7(9): e43810).

As for every useful theory, the MWC framework has since been refined and extended, for instance to encompass the interactions between several regulators, lattices of monomers etc.  I’ll finish by a little advertisement for a conference to celebrate the 50th anniversary of the allosteric regulation


Modelling success stories (1) Hodgkin-Huxley 1952

The model of Hodgkin-Huxley is one of the most (the most?) brilliant examples of computational model explaining quantitatively a living process. In addition, the work involving mathematical modelling, numerical simulation and data-based parametrization, it marks IMHO the starting point of Systems Biology (despite the fact that the name was coined by Bertalanffy in 1928, and the domain really exploded in 1998).

The model provides a mechanistic explanation of the propagation of action potentials in axons, based on the combined behaviours of a system of ionic channels. The model itself is described in a highly cited paper (12635 times according to Google Scholar on Feb 19th 2013; However we know that GS largely underestimate citations of papers published before the “web”):

Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 1952 Aug;117(4):500-544. PDF2

However, this paper is the culmination of a fantastic series of articles published back to back in Journal of Physiology.

  • Hodgkin AL, Huxley AF, Katz B. Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J Physiol. 1952 116(4):424-448
  • Hodgkin AL, Huxley AF. Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J Physiol. 1952 116(4):449-472
  • Hodgkin AL, Huxley AF. The components of membrane conductance in the giant axon of Loligo. J Physiol. 1952 116(4):473-496
  • Hodgkin AL, Huxley AF. The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. J Physiol. 1952 116(4):497-506

In total, we have 126 pages that changed the way we understand how nervous systems, and ultimately our brain, function. The last article is by far the most extraordinary piece of science I personally read.

Part I analyze their experimental results and should be given to every student as a model of scientific reasoning. Their conclusions predict the existence of voltage-sensing ionic channels, different for sodium and potassium, and even the existence of their segment S4, a charged domain sensing the difference of potential and moving accordingly. Note that in 1952, they had absolutely no clues about transmembrane channels, or the nature of the excitable membrane!

In Part II, Hodgkin and Huxley start from the description of an electrical circuit, a natural starting point for them since they recorded electrical properties of the giant squid axon, and progressively derive a model that mechanistically account for each biochemical event, each structural transition of the ionic channels. They even predict the existence of four gates for the channels. Using extremely accurate experimental measurements, they fit the model and determine the values of the different parameters regulating the opening and closing of sodium and potassium channels.


Electrical circuit equivalent of the Hodgkin-Huxley model. The resistors are ionic channels, the capacitance represents the plasma membrane (made of lipids, hence an insulator) and the battery represents the injected current (or voltage, according to the experimental set-up).

Part III puts Humpty Dumpty together again. Hodgkin and Huxley gathered a system of 4 differential equations, 1 for the voltage and 1 for each type of gate, and 6 assignment rule determining the propensity of each gate type to open or close in function of the voltage. Since the unique electrical computer of Cambridge University was out of order, they then simulated the model using a hand-operated machine! The rest is history, and Hodgkin and Huxley won the 1963 Nobel Prize in Physiology or Medicine. Hodgkin-Huxley models are still used in modern multi-compartment models of neuronal systems, for instance used in the Blue Brain Project. More information can be found in the BioModels Database model of the month writen by Melanie Stefan, and on the relevant Wikipedia page.

Coming out of the closet: I like ENCODE results, and I think the “switches” may be functional

It is fair to say that ENCODE has been the big story of 2012 as far as biological research is concerned. There are many reasons for that, including ambition of the project, size of the collaboration, method of publication, amount of press coverage, and surprising findings. The two latter points also started controversies and heated debates that now promise to entertain us in 2013 too. Recently a very peculiarly worded article increased the heat a little, focusing on the criticism of ENCODE’s “80% of the genome is functional” by evolution scientists. For the French readers, a polite discussion of the criticism can be found in Marc Robinson-Rechavi’s blog.

The main criticism can be roughly summarised as: 1) We cannot dissociate biological function from evolution. 2) Evolution is due to selection. 3) Selection results in sequence conservation. 4) ENCODE’s “80% of the genome” is not conserved, therefore it is not functional. Now, I was not involved in ENCODE, and I am not an expert in genomics. However, I am a systems biologist, I know a bit about evolution, and the way most criticisms of ENCODE are expressed itch me a bit. Assumption 3) in particular is problematic for me.

[NB: Something I find quite funny in an ironical sort of way is that initial criticisms went as “we’ve known for years that there was no junk DNA”, see this post by Michael Eisen. And now it goes “they have not shown that DNA was not junk”.]

Let’s leave ENCODE for a moment, we’ll come back to it later. What annoys me the most with some of the vocal criticisms, is the sort of ultra-Dawkinsian position on evolution claiming that it acts solely at the level of the genetic atom (nucleotide or gene according to the context). It may be mostly true for the fundamental bricks of life. If you change the sequence of a core enzyme, the effects will be drastic, and therefore the sequence is very conserved. The more central the process, the higher the conservation (and the lower the redundancy, see my comment on redundancy and evolution, written before the blog era). However, it is not always so true for less central, and more complex (in terms of number of components), systems.

Even if hereditary information is transmitted via discrete pieces of genetic information, evolutionary selection acts at the level of the systems, which is what produce the phenotypes. This is illustrated in the lower panel of the famous Waddington epigenetic landscape.


Selective pressure acts on the surfaces. What is transmitted are the black blocks. The entangled cables between them represent the genotype-phenotype relation (or genotype-phenotype problem if we are more pessimistic than curious). I like this picture very much (while the upper panel is most often shown, in the context of canalization) because IMHO it represents the object of physiology (the surfaces), of genetics (the blocs) and of systems biology (the cables).

Let’s take the example of olfactory receptors (see for example this article). We have hundreds of them encoded by genes distributed in clusters. Roughly, the clusters are pretty conserved, at least in mammals. However within the clusters, frequent gene duplications and loss of function means that there are no strong conservations (see fig 3 of this article). Variability is even seen between chimps and humans. One possible explanation is the way olfaction works. Although some key receptors are associated with given chemical compounds, olfaction seems to work largely on combinatorics. Because a substance can bind several receptors, and the receptors response is integrated by the downstream neuronal network, a few hundreds receptors allow us to discriminate between a fantastic diversity of chemicals. What is selected is the variety of olfactory receptors rather than the receptors themselves. The fact that the olfactory receptors are not conserved does not mean they are not functional. I am pretty confident that if we took out all olfactory receptors of mice, and replaced them with human olfactory receptors, the mice would still be OK (less than the wild-type mice because of the loss of recognition diversity). But if we just take out all the olfactory receptors, the mice would die in nature. Guaranteed.

Now back to ENCODE. I will focus on the “switches”, the very large set of sites that were found to bind transcription factors. For a discussion of the somehow related issue of pervasive transcription (ENCODE’s “60%”), see the recent blog post by GENCODE. Millions of binding sites for transcription factors have been uncovered by ENCODE. They are not always directly linked with the regulation of gene expression, in that the binding of the transcription factor on the site does not physically affect the activity of an RNA polymerase nearby. Therefore, opponents criticize the adjective “functional” for those binding sites. This is where I think they are wrong, or at least a bit rigid. We are not talking about non-specific binding events here, noise due to random stickiness of transcription factors. We are talking about specific binding sites, that just happened not to be always associated with direct transcription events. To say that they are not functional is equivalent to say that the binding of calmodulin to neurogranin is not functional because it does not trigger the opening of a channel, or the activation of an enzyme. Of course it does not (or we think it does not), but the buffering effect changes completely the dynamics of activation for the other targets of calmodulin. The “function” of neurogranin is to bind calmodulin, and do nothing with it.

The potential consequences of the pervasive transcription factor binding on the regulation of nuclear dynamics are enormous. A few millions transcription factor binding sites mean around a few thousands binding sites per transcription factor. Transcription factors are present in the nucleus in fairly low quantities, from dozens to thousands of molecules. The result of having the same number of binding sites than the number of ligands (or more), is called ligand depletion. It is well known of pharmacologists, but less of cell biologists. For striking effects of ligand depletion in the case of cooperative sensors you can read our paper on the topic. For an example closer to gene regulation, see the paper of Matthieu Louis and Nicolas Buchler. So, if the number of binding sites for a transcription factor affects its dynamics, maybe there is a function associated with a number of binding sites. Maybe each binding site is not conserved. Maybe there is no selection pressure for it to be exactly there. But could we not envision that what is selected is the number of transcription factor binding sites, in a situation similar to the olfactory receptors one?

We know that the distribution of chromatin in the nucleus is not at all a plate of spaghetti, with a random location of each gene. There are precisely located transcription factories, where specific genes are targeted (see this paper from Peter Fraser’s group and a recent review). What if those millions of transcription factor binding sites were not randomly located but concentrated in nuclear subdomains? Would it not affect the dynamics of the transcription factors in the nucleus?

I have to confess I have not yet read the ENCODE papers dealing with epigenetic markings. However, would it no be cool if epigenetic marking could mask or reveal hundreds of binding sites in one go? As proposed in our paper on ligand depletion, that would be a possible mechanism to change quickly the dynamic range (the range of active concentration) and the ultrasensitivity of responses to transcription factors.

All that is perhaps science fiction. Perhaps those millions binding sites are randomly located in 3D. Perhaps their availability is not dynamically regulated. But scientific research is pushed a lot by “what if?” questions. Besides, those binding sites exist.

At the end of the day, what I see on one side is an enormous amount of data put out there by enthusiastic scientists for everyone to pick and study further. And what I see on the other side is bitterness, and now anger and rudeness. The proper answer to ENCODE claims lies in ENCODE data. Dear Dan Graur and followers, rather than criticize, with fuzzy arguments and rude words, the statistics of ENCODE, run you own. The data is out there for you.