Geometry in eye of fruit fly suggests energy-morphogenesis relation

The most common organism used for biological modeling is Drosophila melanogaster: the common fruit fly. Because of its minimal care and efficient breeding, it is used in innumerable biological fields such as genetics, physiology, and microbial pathogenesis, to name a few. Mechanical engineers are no strangers to modeling—so naturally, when biologists studying morphogenesis in fruit flies started to consider mechanical factors, they needed someone with a strong mechanical background in soft tissues.

MechSE associate professor Sascha Hilgenfeldt is interested in the geometry, structure, and evolution of foam and soft condensed matter. This specialization lends itself very well to the study of cellular structures and biological tissues, specifically the mechanical forces and statistical trends that affect their growth. Using the eyes and wings of the fruit fly as models, Hilgenfeldt’s work is increasing biology's understanding of morphogenesis: how organisms and their components develop their shape.

"Biologists know a tremendous lot about what proteins are involved in the morphogenesis of each shape," Hilgenfeldt said. "They know when they are expressed, in what sequence, and so on and so forth. But knowing which proteins are being used doesn’t directly relate to the shape."

Hilgenfeldt started by writing an energy functional (the calculated total energy of the system, dependent on the state of the system) for the structure of a fruit fly’s eye. The functional incorporated energy contributions from only two factors: that the barrier between cells was elastic, and that the cells were glued to their neighbors by adhesion molecules. And since a system can only be in equilibrium when the total energy has been minimized, the functional calculated what pattern of shapes would create the lowest possible energy. Impressively, the shapes that required the minimal amount of energy to maintain exactly matched the shapes present in the ordered pattern of a fruit fly’s eye.

"The shape is a consequence of passive energy minimization, and not of active biological processes," Hilgenfeldt said. "That encouraged us, and led us to discover that a lot of morphology can be easily understood with the biology only establishing just a few parameters. This amazingly regular shape was ultimately made from something that was very disordered to start with, less than two days before. And that has huge consequences, because if we were to really understand all the processes during morphogenesis, then we could manipulate the morphogenesis, we could generate tissues, we could tailor tissues that are grown in the lab to purposes that we'd like. These mechanical energy minimizations can really help identify the sequence of certain events during morphogenesis, even if we can only infer them from the outcome."

These ordered shapes can be quantitatively explained by the minimization of energy. However, this model only works in tissue with a regular cellular pattern, such as that of a fruit fly's eye. The tissue of a fruit fly’s wing is disordered, with an irregular pattern that changes from fly to fly. Hilgenfeldt therefore chose to approach the problem from a statistical angle, taking into account the number of neighbors each cell has, and the area of each cell. He found that as the wing develops, the probability distribution of both variables narrows over time.

"This is something that people have observed in a number of systems, not just biological tissue," Hilgenfeldt said. "There is a strong correlation between the distribution width of areas and number of neighbors in this kind of statistical tissue. Nobody has been able to really explain it. What’s important is that we can get analytical solutions for the general appearance of the tissue using only the characteristic width of the area distributions. I don’t need to know anything about the biological processes."

Finding such a universal property is exciting in that explains observations that had previously only been noticed empirically. However, its universality means that it cannot be specific, and therefore cannot give a diagnostic value for an individual pattern. If, for example, one tried to tell the difference between healthy and diseased wing tissue, the correlation between area distribution and neighbor distribution should hold for both.

Hilgenfeldt still thinks there are more answers to be found.

"The ultimate goal is to understand the development from one pattern to another over time, and in practically all cases in biology, this happens over a time scale that is very long compared to mechanical equilibration," Hilgenfeldt said. "These mechanical models actually give definitive answers about the elementary processes from which you can then build the overall morphogenesis. As far as the statistical analysis goes, we’d like to identify better diagnostic measures. And in the course of doing that, hopefully be able to connect back to the mechanics of the system. I think there’s a bridge to be constructed between the statistical analysis and the mechanical analysis, and I hope to pursue that more."