Seeing in Numbers

Statistical mechanics is a field of physics that penetrated chemical and engineering applications in the twentieth century. It is a conceptual mathematical framework that describes and predicts how macroscopic phenomena arise from assemblies of fluctuating and interacting microscopic components. I knew no immunology or immunologists, but I felt like exploring whether these concepts might fruitfully complement experimental research in immunology.

Luckily, the stars seemed aligned for me to try a new problem, for that very fall, I had won an award to spend a semester focused only on research of my choosing, with no teaching or administrative duties. I found myself consuming papers on the immunological synapse, and began learning about immunology in general.

The authors of the paper given to me by Jay described how, after the initial contact between a T cell and an APC, the primary interacting receptors and other molecules shuffle around to form the specific spatial pattern named the immunological synapse (see graphic below). It is comprised of a ring of adhesion molecules that tighten the connection between the cells and enclose a cluster of T cell receptors (TCRs) that bind to MHC-peptide complexes (the ligand) on the APCs. I first worked on the forces that drive synapse formation, but the problem that the field was struggling with at the time was this: what exactly was the function of the synapse? It was first thought that the synapse served to sustain signaling through the TCR, and thereby enable T cell activation and differentiation. But a subsequent study led by Andrey Shaw reported that there were no active signaling molecules at the center of the synapse where the TCRs were clustered.³ If not to transduce signaling, it was unclear what the purpose of the synapse might be.

My lab started working on a number of computer simulations that simultaneously accounted for different possible biochemical events and how they may be influenced by the synapse. The results suggested an explanation for the observed paucity of active signaling molecules in the center. When a TCR interacts with a matching MHC-peptide ligand, kinases within the lymphocyte are recruited, phosphorylating multiple sites on the cytoplasmic tail of the TCR. More phosphorylated receptors transduce signals more efficiently, but they also become substrates for ubiquitination and receptor degradation. Our studies suggested that, for the typical experimental situation in which high doses of ligand that bind strongly to the TCR are used, the TCRs are efficiently phosphorylated before they are clustered in the center. This cluster of receptors, which forms after the initial signals have been relayed, makes ubiquitination of the phosphorylated tails more efficient, which in turn promotes receptor degradation and attenuation of strong signals. It explained why Shaw’s group had seen no active signaling molecules within the synapse. I spoke to Andrey about these findings, and with help from Mike Dustin’s lab, he designed a number of experiments that supported the mathematical prediction. Studies conducted by others also concluded that one function of the synapse was to attenuate TCR signaling, important because too much TCR signaling would trigger cell death in the lymphocyte. But our work with Andrey (now a dear friend) suggested that the situation is more subtle. When ligands bind weakly to the TCR, receptor phosphorylation may not occur efficiently unless the receptors are clustered at a synapse. In these cases, clustering can first promote receptor phosphorylation and then signal attenuation by ubiquitination.⁴

The immune system is an area of biology that can especially benefit from computational modeling because it is a complex evolving system that does not necessarily behave in an obviously linear pattern. Mathematical modeling can lead to insights into mechanisms underlying these nonintuitive behaviors that can then be tested in the lab.

umping into a new field as a principal investigator also made vivid the many vestigial barriers to interdisciplinary research. An amusing example concerns my application for membership to the American Association of Immunologists (AAI). The AAI had a rule that you must have at least one “first author” paper in immunology and, as I came to immunology as a full professor, I was listed as a corresponding author. My application was rejected! This rule has since been changed. Today I am a card-carrying AAI member, and I feel less like an imposter.

In contrast to such barriers associated with organizations that support scientific research, immunologists have been wonderfully receptive to learning how the language of math and physics might be used to inform and predict immune interactions. I first experienced this when I presented my first paper in immunology at a Keystone conference poster session, and Mike introduced me to several immunologists. Mark Davis from Stanford University came by to see my poster and our discussions led to collaborations that continue today. He has been a great teacher as I have continued to learn about this field. Since then, I have had similar experiences with many other immunologists.

Immunologists first thought that the function of the synapse—the receptors at the interface of a T cell and an antigen-presenting cell (APC)—was to keep the two cells in contact long enough to phosphorylate the T cell receptor’s (TCR) cytoplasmic tails and activate the cell. Our studies show that there is more subtlety involved. When the match between the receptor and ligand is strong, the TCRs are phosphorylated even before the synapse is fully formed (top left). In this situation, the formation of the synapse acts to cluster the activated TCR cytoplasmic tails so that ubiquitin can degrade the signal and prevent the T cell’s over-activation and death (top right). But when the signal is weak (bottom), the synapse helps promote signaling followed by ubiquitination.

In 2006, I was invited to give my first talk at a Keystone conference focused on immunology. The night before my presentation, I heard a talk by Arthur Weiss about his research on signaling cascades involved in T cell activation. His fascinating biochemical observations brought to mind a physical phenomenon called hysteresis, which is characterized by a system’s response being dependent on the last action exerted on it. Iron, for example, will remain magnetized—its dipoles aligned in one direction—even after the magnet is removed, demonstrating a kind of memory of the last action exerted on it. I wondered if T cell signaling cascades could exhibit hysteresis. If so, this might reveal new patterns of behavior predictable by mathematical formulas. Back in my hotel room over chicken quesadillas, I worked out some rough calculations using differential equations that describe the biochemical reactions involved to explore my hypothesis. I was delighted to see that hysteresis might occur!

In his talk, Art had described a series of signaling events that occur after a T cell receptor binds a peptide MHC and its cytoplasmic tails are phosphorylated. One consequence is the activation of Ras—a well known protein product of an oncogene involved in cell growth and differentiation and a critical component of many signaling pathways. Active Ras proteins, in turn, activate a variety of signaling pathways that ultimately lead to gene transcription events necessary for T cell activation. Lymphocytes have a Ras activator protein called SOS. Art described his work on the interplay between SOS and another Ras activator, and noted that John Kuriyan’s lab had shown that SOS is subject to positive feedback regulation. SOS activates Ras by binding inactive Ras proteins at a catalytic site. If an activated Ras protein binds to another allosteric site on SOS, the rate at which SOS activates other Ras molecules goes up approximately 100-fold. But activated proteins don’t remain in the cytoplasm forever. They are degraded by ever-present Ras deactivator proteins, like RasGAP. My calculations suggested that, even after stimulus was withdrawn and free active Ras proteins were degraded by RasGAPs, active Ras molecules could stay bound to SOS, thereby maintaining a more active catalyst than if the cell had not been previously stimulated—thus demonstrating a short-term memory of activation or hysteresis (see graphic below).

When I talked to Art about his work the next day, he was amazingly receptive to learning about my ideas on what must have been for him an obscure physical property. Sharing a cab back to the airport after the conference, we began to solidify our plans for collaboration. Over the next 2 years, our labs took the iterative approach of making predictions by theoretical models, designing experimental tests, refinements to the model, and further tests; efforts were led by Jayajit Das in my lab and Jeroen Roose in Art’s lab.

Our modeling and experimental studies revealed that feedback regulation of SOS’s enzymatic activity plays an important role in mediating two aspects of T cell signaling: hysteresis and digital signaling. If a population of T cells is stimulated very weakly, the number of activated downstream signaling molecules (like Ras) is maintained at a weak basal level—usually insufficient to activate the cell. Instead of responding in a continuous or “analog” fashion, in which an increase in T cell stimulation would result in a commensurate increase in level of activation, Ron Germain’s lab had observed that T cells respond in a way that could be described as “digital.” Above a threshold stimulus level, T cells split into two subpopulations, one that turns on a large number of active signaling molecules (like Ras), while the other maintains basal signaling levels, insufficient for activation. Our work suggests that positive feedback regulation of SOS may underlie this “on” or “off” response.

A T cell that receives a weak stimulus would not activate sufficient amounts of Ras and other downstream signaling components to activate the cell, and would remain “off.” However, if the cell is first stimulated by a strong signal, even after the stimulus is removed, for a finite duration, SOS proteins would retain enough active Ras in their allosteric pockets to remain potent Ras activators. This “memory” of previous stimulation would now allow subsequent weak stimuli (which do not activate Ras in previously unstimulated cells) to robustly activate Ras. This would result in the build-up of downstream signaling molecules that cause T cell activation. If one waits too long after the first strong stimulation, however, active Ras levels fall below a sharp threshold, and previously stimulated cells cannot be robustly activated. This short-term memory of past encounters with strong stimulus, hysteresis, occurs because of the nonlinearities in the positive feedback regulation. Our studies suggest that under physiological doses of antigen, T cells may be able to sum up signals from multiple interrupted encounters with APCs bearing low doses of antigen.⁵ This finding may have implications for how T cells sense antigen during natural infections and vaccination. My collaborators and I continue to explore these and other phenomena that determine how T cell signaling is regulated.

s I learned more about T cell biology, I started to yearn to work on other aspects of it, and felt that methods rooted in statistical mechanics could help elucidate aspects of processes that span a diverse range, from molecules and cells to organs to the organism. Moving to MIT and the Boston area—which had a higher density of diverse medical scientists—enhanced my desire to work on problems other than T cell signaling.

One of the projects I have worked on has subsequently sparked collaborations with clinicians, but it started with a basic question that Herman Eisen raised: How do T cells recognize the huge range of foreign antigens while also exhibiting exquisite specificity for particular antigens? The precursors of T cells are generated in the bone marrow, each bearing a unique TCR with a slightly different amino acid sequence on the head of the receptor. Creating an army of diverse TCRs gives the immune system the best chance to match sequences with the small peptides derived from the huge and evolving array of foreign invaders it may encounter over the course of a lifetime. But because this process of generating TCR sequences is random, some of the sequences will be able to bind to self-antigen. In order for the body to reject the T cells which have a strong affinity for self peptides, immature T cells travel to the thymus, to be “educated” by antigen-presenting cells that display self peptides bound to MHC proteins. All T cells that bind strongly to these self-MHC-peptide molecules commit suicide (negative selection). Furthermore, a T cell that emerges into the blood to do battle with pathogens must also bind weakly to at least one self-MHC-peptide (positive selection). A faulty process of “self”education is thought to be one trigger of autoimmune diseases.

One big puzzle in this area is that recognition—or a TCR binding strongly enough to a foreign MHC-peptide to activate the T cell—can be both highly specific and at the same time flexible, or degenerate. A small change in a peptide recognized by a TCR will likely abrogate recognition. Paradoxically, a single TCR can recognize multiple unrelated peptides. Recently, we have helped understand how the education process in the thymus can tune TCRs to be both highly specific and degenerate.

I was inspired by the work of Eric Huseby, Pippa Marrack, and John Kappler, showing that T cells would become far more cross-reactive, or degenerate, in their binding if they were trained against only one self peptide in the thymus. T cell selection is like trying to attain perfect imperfection: in order to be allowed to circulate in the body, each T cell must never bind strongly—with perfect matching—to any MHC-self peptide it encounters. The situation brought to my mind a phenomenon characteristic of “spin glasses.” Here magnetic impurities in materials have to align in random orientations with other magnetic impurities in order to be energetically favorable, rather than neatly orienting in the same direction

When a T cell receives a signal from an APC, it activates Ras proteins via a Ras activator enzyme called SOS. We found that SOS has the ability to retain the “memory” of its past encounter—a mathematical concept called hysteresis—by holding activated Ras bound in its allosteric site while free Ras becomes deactivated over time. This ability to “remember” allows a T cell to robustly activate Ras from a subsequent weak stimulation, following at least one stronger stimulus.

Along with my physics colleague, Mehran Kardar, and my student, Andrej Košmrlj, I developed an in silico model of thymic selection and used mathematical ideas from spin glass physics to understand our results. Our thymic selection model revealed that avoiding strong binding to a diversity of self-peptides during education in the thymus resulted in mature T cells with peptide contact residues enriched in amino acids that interacted weakly with other amino acids. TCRs with peptide contact residues that were strongly charged or had very flexible side chains were likely eliminated because they had a high probability of encountering a self-peptide with which it would have strong interactions, resulting in suicide. The weakly binding amino acids on peptide contact residues of mature T cells mediated recognition of antigenic peptides via a number of moderate to weak interactions, each contact being likely to contribute significantly to recognition. Mutations to any one amino acid on the peptide thus can cause abrogation of recognition—that is, specificity. But, while one mutation can abrogate recognition, if we make a number of mutations that enable a TCR to establish a number of weak interactions that sum up to the binding energy required for recognition, a TCR will effectively recognize a different peptide, giving the system the property of degeneracy.⁶ Rather than Emil Fisher’s “lock and key” metaphor for enzyme-substrate specificity, we propose a more statistical model for TCR recognition of antigen.

These findings sparked my collaboration with Bruce Walker and his colleagues at Massachusetts General Hospital (MGH). I am very excited about exploring the implications for human immune responses to natural infections such as HIV. The ultimate goal of this line of multi-disciplinary research being pursued at the Ragon Institute of MGH, MIT, and Harvard is to inform design of better vaccines.

Although my past research on polymers and catalysts was interesting, my work applying approaches from the physical and engineering sciences to immunology has captured my imagination and is the most personally satisfying work I’ve ever done. I look forward to many more years at the crossroad of these disciplines.

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Arup K. Chakraborty is Robert T. Haslam Professor of Chemical Engineering, Chemistry, and Biological Engineering at the Massachusetts Institute of Technology. His multidisciplinary computational immunology research group works on various aspects of immunology in close collaboration with experimental labs.