The advantage of such parsimony is that it is easier to analyze how the interaction between negative and positive feedback regulates model behavior. The biological wood starts to emerge from the molecular trees, much as it did for Heinrich and Rapoport in the previous example. But the goal here also involves the interpretation of quantitative experimental data.
Altan-Bonnet and Germain opted instead for a detailed model based on the known biochemistry.
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Their model has around dynamical variables. Only the core module is described in the main paper, with the remaining nine modules consigned to the Supplementary Graveyard. The tension between parsimony and detail runs through systems biology like a fault line. To some, and particularly to experimentalists, detail is verisimilitude. The more a model looks like reality, the more it might tell us about reality.
The devil is in the details. But we never bother ourselves with all the details. All those phosphorylation sites? We are always simplifying—ignoring what we think is irrelevant—or abstracting—replacing something complicated by some higher-level entity that is easier to grasp. It is impossible to think about molecular systems without such strategies: it is just that experimentalists and mathematicians do it differently and with different motivations. There is much to learn on both sides, for mathematicians about the hidden assumptions that guide experimental thinking, often so deeply buried as to require psychoanalysis to elicit, and for experimentalists about the power of abstraction and its ability to offer a new language in which to think.
We are in the infancy of learning how to learn from each other. The principal disadvantage of a biologically detailed model is the attendant parameter problem. Parameter values are usually estimated by fitting the model to experimental data. What von Neumann meant is that a model with too many parameters is hard to falsify. It can fit almost any data and what explanatory power it might have may only be an accident of the particular parameter values that emerge from the fitting procedure. Judging from some of the literature, we seem to forget that a model does not predict the data to which it is fitted: the model is chosen to fit them.
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In disciplines where fitting is a professional necessity, such as X-ray crystallography, it is standard practice to fit to a training data set and to falsify the model, once it is fitted, on whether or not it predicts what is important [ 40 ]. In other words, do not fit what you want to explain! Remarkably, Altan-Bonnet and Germain sidestepped these problems by not fitting their model at all.
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They adopted the same tactic as Heinrich and Rapoport and set many similar parameters to the same value, leaving a relatively small number of free parameters. Biological detail was balanced by parametric parsimony. The free parameters were then heroically estimated in independent experiments. I am told that every model parameter was constrained, although this is not at all clear from the paper.
We see that the published model emerged through a cycle of falsification, although here it is the model that falsifies the interpretation of population-averaged data, reminding us yet again that the mean may not be representative of the distribution. With the measured parameter values, the model exhibits a sharp threshold at a pMHC-TCR lifetime of about 3 seconds, above which a few pMHCs 10 to are sufficient to trigger full downstream activation of ERK in 3 minutes.
Lifetimes below the threshold exhibit a hierarchy of responses, with those close to the threshold triggering activation only with much larger amounts of pMHCs , , while those further below the threshold are squelched by the negative feedback without ERK activation. This accounts well for the specificity, sensitivity and speed of T-cell discrimination but the authors went further. They interrogated the fitted model to make predictions about issues such as antagonism and tunability and they confirmed these with new experiments [ 29 ].
The model was repeatedly forced to put its falsifiability on the line.
In doing so, the boundary of its explanatory power was reached: it could not account for the delay in ERK activation with very weak ligands and the authors explicitly pointed this out. This should be the accepted practice; it is the equivalent of a negative control in an experiment. A model that explains everything, explains nothing. Even von Neumann might have approved. To be so successful, a detailed model relies on a powerful experimental platform. The RMA-S target cells were derived from a lymphoma that was mutagenized to be deficient in antigen processing, so that the cells present only exogenously supplied peptides on MHCs.
T-cell activation was measured by flow cytometry with a phospho-specific antibody to activated ERK.
In this way, calibrated amounts of chosen peptides can be presented on MHCs to a single type of TCR, much of the molecular and cellular heterogeneity can be controlled and quantitative data obtained at the single-cell level. Such exceptional experimental capabilities are not always available in other biological contexts. Animals exhibit repetitive anatomical structures, like the spinal column and its attendant array of ribs and muscles in vertebrates and the multiple body segments carrying wings, halteres and legs in arthropods like Drosophila.
During vertebrate development, repetitive structures form sequentially over time. In the mid s, the developmental biologist Jonathan Cooke and the mathematician Chris Zeeman suggested that the successive formation of somites bilateral blocks of mesodermal tissue on either side of the neural tube—see Figure 3 might be driven by a cell-autonomous clock, which progressively initiates somite formation in an anterior to posterior sequence as if in a wavefront [ 41 ].
They were led to this clock-and-wavefront model in an attempt to explain the remarkable consistency of somite number within a species, despite substantial variation in embryo sizes at the onset of somitogenesis [ 42 ]. In the absence of molecular details, which were beyond reach at the time, their idea fell on stony ground. His laboratory showed, using fluorescent in situ hybridization to mRNA in tissue, that the gene c-hairy1 exhibits oscillatory mRNA expression with a period of 90 minutes, exactly the time required to form one somite [ 43 ].
Such oscillatory genes in somite formation were termed cyclic genes. The somitogenesis clock. Bottom left: Potential auto-regulatory mechanisms in the zebrafish, taken from [ 47, Figure three A,B ]. In the upper mechanism, the Her1 protein dimerizes before repressing its own transcription. In the lower mechanism, Her1 and Her7 form a heterodimer, which represses transcription of both genes, which occur close to each other but are transcribed in opposite directions. Explicit transcription and translation delays are shown, which are incorporated in the corresponding models. Bottom right: Mouse embryos stained by in situ hybridization for Uncx4.
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As to the mechanism of the oscillation, negative feedback of a protein on its own gene was known to be a feature of other oscillators [ 45 ] and some cyclic genes, like hes7 in the mouse, were found to exhibit this property. Negative feedback is usually associated with homeostasis—with restoring a system after perturbation—but, as engineers know all too well, it can bring with it the seeds of instability and oscillation [ 46 ].
However, Palmeirim et al. It remained unclear how the clock worked. The developmental biologist Julian Lewis tried to resolve this question in the zebrafish with the help of a mathematical model [ 47 ].
Zebrafish have a very short somite-formation period of 30 minutes, suggesting that evolutionary tinkering may have led to a less elaborate oscillator than in other animals. The HES family genes her1 and her7 were known to exhibit oscillations and there was some evidence for negative auto-regulation.
Lewis opted for the most parsimonious of models to formalize negative auto-regulation of her1 and her7 on themselves, as informally depicted in Figure 3. However, he made one critical addition by explicitly incorporating the time delays in transcription and translation. Time delay in a negative feedback loop is one feature that promotes oscillation, the other being the strength of the negative feedback.
Indeed, there seems to be a trade-off between these features: the more delay, the less strong the feedback has to be for oscillation to occur [ 48 ]. The idea must have been in the air because Jensen et al. The model parameters, including the time delays, were all estimated on the basis of reasonable choices for her1 and her7 , taking into account, for instance, the intronic structure of the genes to estimate transcriptional time delays. Nothing was fitted. With the estimated values, the models showed sustained periodic oscillations.
A pure Her7 oscillator with homodimerization of Her7 prior to DNA binding which determines the strength of the repression had a period of 30 minutes. As with the Heinrich—Rapoport model, there is no data but much biology. What is achieved is the demonstration that a simple auto-regulatory loop can plausibly yield sustained oscillations of the right period. This suggests a different interpretation of Palmeirim et al. The period for somite formation in the mouse is minutes and evidence had implicated the mouse hes7 gene as part of the clock mechanism.
Assuming a Hes7 half-life of 20 minutes against a measured half-life of The model also showed that if Hes7 was stabilized slightly to have a half-life only 10 minutes longer, then the clock broke: the oscillations were no longer sustained but damped out after the first three or four peaks of expression [ 52, Figure six B ]. Hirata et al. The K14R mutant was found to repress hes7 transcription to the same extent as the wild type but to have an increased half-life of 30 minutes.
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Further work from the Kageyama laboratory, as well as by others, has explored the role of introns in determining the transcriptional delays in the somitogenesis clock, leading to experiments in transgenic mice that again beautifully confirm the predictions of the Lewis model [ 53 — 55 ]. These results strongly suggest the critical role of delays in breaking the clock but it remains of interest to know the developmental consequences of a working clock with a different period to the wild type [ 56 ].
here I cannot think of any other model that can so accurately predict what happens in re-engineered mice. On closer examination, however, there is something distinctly spooky about it. If mouse pre-somitic mesodermal cells are dissociated in culture, individual cells show repetitive peaks of expression of cyclic genes but with great variability in amplitude and period [ 57 ].
In isolation, the clock is noisy and unsynchronized, nothing like the beautiful regularity that is observed in the intact tissue. The simple Lewis model can be made much more detailed to allow for such things as stochasticity in gene expression, additional feedback and cell-to-cell communication by signaling pathways, which can serve to synchronize and entrain individual oscillators [ 47 , 58 — 60 ]. A more abstract approach can also be taken, in which emergent regularity is seen to arise when noisy oscillators interact through time-delayed couplings [ 61 , 62 ].
The thing is, none of this molecular complexity is present in the Lewis model. Yet, it describes what happens in the mouse with remarkable accuracy. The microscopic complexity seems to have conspired to produce something beautifully simple at the macroscopic level.
How does the Lewis model emerge in the tissue from the molecular complexity within? It is as if we are seeing a tantalizing glimpse of some future science whose concepts and methods remain barely visible to us in the present. Every time I think about it, the hairs on the back of my neck stand up. A mathematical model is a logical machine for converting assumptions into conclusions.
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