Genes on the brink of chaos: what chaos
It had been a long time since I had written prose. I sometimes write poetry in my spare time, but unfortunately we end up losing some of our habits due to our interests. As you read this and ponder why I wrote such an intro to a research paper, some of the billions of neurons in your brain are resting at intervals, while others are regularly interacting with nearby neurons. . They send regulatory instructions to some of your organs while you do not feel the slightest change. The same mechanism could repeat itself continually in the microscopic world as you read this, unconsciously sipping your tea or coffee, as your lungs continue to inflate and deflate. From inside the brain of our cells, the nucleus, instructions are constantly sent to the ribosomes and other organelles.
Genes, which reside in the DNA stored by the nucleus, send the recipe for proteins that the cell may need for its activities to the ribosomes in the form of mRNA. In this sense, while the nucleus serves as the cellular brain, the building blocks for performing these functions are the genes rather than the neurons. The activation of a gene may depend on other genes due to the relationships between these genes. The activation of a gene can inhibit the functioning of another gene in certain cases, as well as promote it. All of these binary relationships form the gene regulatory networks, similar to neural networks as a whole.
So, when our bodies are made up of trillions of cells that carry the same genes in their DNA, how come each one specializes to perform different functions? How do heart cells, which constantly contract and relax, and pancreatic cells, which produce digestive enzymes, differentiate and perform very different functions despite having the same genetic material? At this stage, gene regulatory networks turn on genes needed to perform certain functions, and render some dysfunctional, with the help of external signals and cell-to-cell communication. These networks also alter the activities of genes in response to changes that occur inside or outside cells, managing the process of cell adaptation to various environmental conditions. These changes can come from sugar or oxygen concentration levels in intercellular fluid, as well as signals from other cells and the brain itself. In these networks, each gene can be affected by a group of genes or some external factors. Thus, the functioning of a gene is determined by these influencing factors, and each gene can exist in two different states: “active” or “passive”.
Depending on the function of a cell, some of its genes must always be kept away, some must be constantly active, and some must become active from time to time. Therefore, genes that should be consistently functional or non-functional must be insensitive to changes, while other genes may have varying sensitivities depending on their rate of work. A dysfunction of the networks or a mutation can lead to specific diseases. For example, leukemia occurs when there is a defect in the functioning of genes that prevent the division of cells, which should be constantly active under normal conditions. Therefore, examining gene sensitivity levels is important for understanding the functions of gene networks and cells in general, and for examining intracellular gene dynamics.
When we look at average sensitivity levels in gene networks, we see an interesting picture; sensitivity values lie on the boundary between chaotic and stable. The famous biologist S. Kauffman used the “chaos edge” argument to explain this behavior of gene networks that oscillate between chaos and stability. This argument suggests that biological systems should be neither too chaotic (to function reliably) nor too stable (to be able to adapt to change). However, there is more to the behavior of the gene network than this argument alone.
When we compare the average activation rates and sensitivity levels of genes in gene network models, we encounter a pattern that no one has noticed before. Well, could you notice this interesting pattern?
As we mentioned earlier, genes that are always active or always non-functional are not too sensitive. However, the main thing to notice is that the sensitivity values always stay above a certain curve. So is it a biological requirement for gene networks to work, or is it a requirement of the mathematical definition of sentience?
At this point, biology continues to provide more clues. The “channeling” effect that genes have on other genes has long been a subject of debate among researchers. This characteristic expresses the determining effect of a specific gene on the activity of another gene. For example, if gene B is deactivated in all cases where gene A is active, it means that gene A channels gene B to become non-functional. In a more restrictive version of this phenomenon, called “nested channeling”, all genes that affect a certain gene have a say in the activity of that gene.in a hierarchical order. Our observations show that an intense nested channeling effect is found in the gene networks. When we compare the activity and susceptibility of genes alone with this particular condition, the stones begin to set in. The genes least responsive to their activity levels should have this nested feature.
When examining this phenomenon mathematically, it is not possible to go under the curve created by those with nested structures.
This curve, which determines the minimum sensitivity, has a fractal structure. In the complexity of biology, it is not uncommon to see such a degree of order and structure. The pyramidal structure of a Romanesco cauliflower and the capillaries surrounding our lungs are just a few of the fractal structures we encounter in living organisms. This curve is also known as the “Blancmange” curve, which was proposed by Teiji Takagi in 1901 as a continuous function at any point, and not differentiable at any point (no tangent can be drawn at any point). The Blancmange curve, which was only known with this feature for a hundred years, now also acquires biological significance.
After these observations, even though Kauffman’s “edge of chaos” argument and minimal gene sensitivity in gene networks seem to contradict each other at first glance, they actually agree quite well. According to Kauffman, gene networks should neither be too stable nor too chaotic, but our observations show that individual genes have the most stable structure possible relative to their average activity levels. The characteristics of individual genes can lead to different observations in the networks formed by the genes. Just like how simple movements of migrating birds in relation to each other create a dance performance.
In summary, we have discovered a hidden fractal pattern within the complex interaction networks of cell biology. This fractal boundary acts as an invisible wall, pushing organisms towards the frontier of order and chaos, most suitable for living systems. Our findings were recently published in Physical examination letters, the most prestigious journal in the field of physics and mathematics. We were pleasantly surprised to learn that it was also chosen for “Editor’s Pick”, a list that highlights about one in seven letters because of its importance or special appeal.
As we draw inspiration from nature in the world of design and engineering, biological observations are also the engine of pure scientific progress. Our study provides deeper insight into the interplay between mathematics and biology and informs further studies on both fronts. We are currently continuing our investigations on the theoretical level by examining the cases where the nested structure above is sometimes broken. Now we continue our studies with the theoretical examination of the cases where this nested structure follows an on-and-off pattern and the sensitivity levels of these networks.
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