Some Challenges of Biological Modeling

If modeling one small aspect of one cell is this difficult, how much more difficult is it to model whole cells and organisms?

While the biological literature is full of data / knowledge about how cells and organisms work, we remain far from true understanding- the kind of understanding that would allow computer modeling of their processes. This is both a problem of the kind of data, which is largely qualitative and descriptive, and also of amount- that countless processes and enzymes have never had their detailed characteristics evaluated. In the human genome, I would estimate that roughly half its genes have only been described (if at all) in the most rudimentary way, typically by loose analogy to similar ones. And the rest, when studied more closely, present all sorts of other interesting issues that deflect researchers from core data like their enzymatic rate constants and binding constants to other proteins, as might occur under a plethora of different modification, expression, and other regulatory conditions. 

Then how do we get to usable models of cellular activities? Typically, a lot of guessing is involved, to make anything that approaches a computer model. A recent paper offered a novel way to go down this path, which was to ignore all the rate constants and even interactions, and just focus on the measurements we can make more conveniently- whole metabolome assessments. These are experiments where mass spectrometry is used to evaluate the level of all the smaller chemicals in a cell. If such levels are known, perhaps at a few different conditions, then, these authors argue, we can derive models of their mutual regulation- disregarding all the details and just establishing that some sort of feedback system among these metabolic chemicals must exist to keep them at the observed concentrations.

Their experimental subject is a relatively understandable, but by no means simple, system- the management of iron concentrations in yeast cells. Iron is quite toxic, so keeping it at controlled concentrations and in various carefully-constructed complexes is important for any cell. It is used to make heme, which functions not only in hemoglobin, but in several core respiratory enzymes of mitochondria. It also gets placed into iron-sulfur clusters, which are used even more widely, in respiratory enzymes, in the DNA replication, transcription, protein synthesis, and iron assimilation machineries. It is iron's strong and flexible redox chemistry (and its ancient abundance in the rocks and fluids life evolved with) that make it essential as well as dangerous.

Author's model for iron use and regulation in yeast cells. Outside is on left, cytoplasm is blue, vacuole is green, and mitochondrion is yellow. See text below for abbreviations and description. O2 stands for the oxygen  molecule. The various rate constants R refer to the transition between each state or location.

Iron is imported from outside and forms a pool of free iron in the cytoplasm (FC, in the diagram above). From there, it can be stored into membrane-bound vacuoles (F2, F3), or imported to the mitochondria (FM), where it is corporated into iron-sulfur clusters and heme (FS). Some of the mitochondrially assembled iron-sulfur clusters are exported back out to the cytoplasm to be integrated to a variety of proteins there (CIA). This is indeed one of the most essential roles of mitochondria- needed even if metabolic respiration is for some reason not needed (in hypoxic or anaerobic conditions). If there is a dramatic overload of iron, it can build up as rust particles in the mitochondria (MP). And finally, the iron-sulfur complexes contribute to respiration of oxygen in mitochondria, and thus influence the respiration rate of the whole cell.

The task these authors set themselves was to derive a regulatory scheme using only the elements shown above, in combination with known levels of all the metabolites, under the conditions of 1) normal levels of iron, 2) low iron, and 3) a mutant condition- a defect in the yeast gene YFG1, which binds iron inside mitochondria and participates in iron-sulfur cluster assembly. A slew of differential equations later, and selection through millions of possible regulatory circuits, and they come up with the one shown above, where the red lines/arrows indicate positive regulation, and the red lines ending with bars indicate repression. The latter is typically feedback repression, such as of the import of iron, repressed by the amount already in the cell, in the FC pool. 

They show that this model provides accurate control of iron levels at all the various points, with stable behavior, no singularities or wobbling, and the expected responses to the various conditions. In low iron, the vacuole is emptied of iron, and in the mutant case, iron nanoparticles (MP) accumulate in the mitochondrion, due in part to excess amounts of oxygen admitted to the mitochondrial matrix, which in turn is due to defects in metabolic respiration caused by a lack of iron-sulfur clusters. What seemed so simple at the outset does have quite a few wrinkles!

The authors present their best regulatory scheme, selected from among millions, which provides accurate metabolite control in simulation, as shown by key transitions between conditions as shown here, one line per molecular species. See text and image above for abbreviations.

But note that none of this is actually biological. There are no transcription regulators, such as the AFT1/2 proteins known to regulate a large set of iron assimilation genes. There are no enzymes explicitly cited, and no other regulatory mechanisms like protein modifications, protein disposal, etc. Nor does the cytosolic level of iron actually regulate the import machinery- that is done by the level of iron-sulfur clusters in the mitochondria, as sensed by the AFT regulators, among other mechanisms.

Thus it is not all clear what work like this has to offer. It takes the known concentrations of metabolites (which can be ascertained in bulk) to create a toy system that accurately reproduces a very restricted set of variations, limited to what the researchers could assess elsewhere, in lab experiments. It does not inform the biology of what is going on, since it is not based on the biology, and clearly even contravenes it. It does not inform diseases associated with iron metabolism- in this case Friedreich's ataxia which is caused in humans by a gene related to YFH1- because again it is not biologically based. Knowing where some regulatory events might occur in theory, as one could have done almost as well (if not quantitatively!) on a cocktail napkin, is of little help when drugs need to be made against actual enzymes and actual regulators. It is a classic case of looking under the streetlight- working with the data one has, rather than the data one needs to do something useful.

"Like most ODE (ordinary differential equation)-based biochemical models, sufficient kinetic information was unavailable to solve the system rigorously and uniquely, whereas substantial concentration data were available. Relying on concentrations of cellular components increasingly makes sense because such quantitative concentration determinations are becoming increasingly available due to mass-spectrometry-based proteomic and metabolomics studies. In contrast, determining kinetic parameters experimentally for individual biochemical reactions remain an arduous task." ...

"The actual biochemical mechanisms by which gene expression levels are controlled were either too complicated to be employed in autoregulation, or they were unknown. Thus, we decided to augment every regulatable reaction using soft Heaviside functions as surrogate regulatory systems." ...

"We caution that applying the same strategy for selecting viable autoregulatory mechanisms will become increasing difficult computationally as the complexity of models increases."

But the larger point that motivated a review of this paper is the challenge of modeling a system so small as to be almost infinitesimal in the larger scheme of biology. If dedicated modelers, as this laboratory is, dispair of getting the data they need for even such a modest system, (indeed, the mitochondrial iron and sulfur-containing signaling compound that mediates repression of the AFT regulators is still referred to in the literature as "X-S"), then things are bleak indeed for the prospect of modeling higher levels of biology, such as whole cells. Unknowns are unfortunately gaping all over the place. As has been mentioned a few times, molecular biologists tend to think in cartoons, simplifying the relations they deal with to the bare minimum. Getting beyond that is going to take another few quantum leaps in data- the vaunted "omics" revolutions. It will also take better interpolation methods (dare one invoke AI?) that use all the available scraps of biology, not just mathematics, in a Bayesian ratchet that provides iteratively better models. 

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