Biological structure exists and interacts in space. Space is 3D, although from the perspective of biological organisms not Euclidean, because gravity acts in one direction on the surface of the earth, and therefore, the degrees of mobility of and interactions between terrestrial organisms are often constrained to something more like 2D. On the scale of cells, this plays a more minor role. Nevertheless, one of the most important structures, the DNA is arranged in a one-dimensional manner. Why is this so? It is not 3D, because the interior of a 3D object is not accessible for readouts or copying. It is not 2D, because a linear structure is better adapted to sequential, temporal processing. That in turn is needed because there are bottlenecks like the ribosome where polypeptides are assembled. Likewise, replication seems to be less error prone and more energy efficient when arranged in a sequential, temporal manner instead of taking place simultaneously, like in a Xerox machine. When replication is carried out sequentially, the same copying molecules and structures can be used repeatedly. Of course, both the DNA itself and also its products, the polypeptides then acquire a 3D shape. For the polypeptides that constitute proteins, this is essential for their biological functions, because in that way particular motives can be exposed to interactions with other substances or shielded from such interactions. For the DNA, the spatial arrangement is important for the regulation of gene transcription, as emphasized by Boi (2011). For instance, via a suitable spatial organization, genomic regions that should be simultaneously transcribed can be brought into spatial proximity, even if their intrinsic linear distance on the DNA could be quite large, thereby facilitating coregulation, as originally proposed by Képès and Vaillant (2003), although their original solenoid model may have been too simple. The spatial organization of the DNA is certainly not as erratic as it originally looked, nor as regular as proposed in the first models, but very carefully orchestrated by specific proteins.
For the RNA, which is not only an intermediate between the DNA and proteins, but the crucial instance for regulation and processing (as well as for a host of catalytic roles), the secondary structure is important, achieved by pairwise bonds between complementary nucleotides in a linear sequence. Much of the processing is regulated by interactions with specific proteins, and in Jost and Scherrer (2014), a combinatorial code has been proposed. In turn, RNA molecules can also function as scaffolds for bringing specific groups of proteins together to induce their functional interaction. For that role, a 2D structure seems to be the most appropriate.
A caricature might then say that we proceed from the 1D DNA (information storage) via the 2D RNA (regulation and processing) to the 3D proteome (cellular function). Of course, as discussed, the DNA is organized in 3D, and RNAs and proteins not only have a 3D shape, but also interact in 3D.
The latter point indicates that 3D geometry is important not only for single structures, like proteins or the DNA, but also for the interaction between structures. It facilitates and constrains interactions at the same time. In 3D, substances can find each other more easily than in higher dimensions, but there are also constraints for simultaneous interactions (see Bailly and Longo 2011, p.122, on this point). If some substance occupies a place in space, that place is no longer accessible to others. The effects may not always be easy to access. Let us consider the example of the toponome project of Walter Schubert, Andreas Dress and their collaborators (Schubert et al. 2006). By repeated staining and bleaching of a cell slice, they can record the positions of about 100 proteins in that slice. In particular, one then has data about the colocalization of proteins. These data can be arranged in a simplicial complex. The vertices of that simplicial complex stand for the various proteins. Two vertices are connected by an edge if the two corresponding proteins frequently occur in neighboring positions. Here, one can set some threshold, how often those proteins should occur together in order to speak of cooccurrence and introduce the corresponding edge. Similarly, one inserts a two-dimensional simplex, that is, a triangle with three vertices, when all three corresponding proteins frequently occur together, and not only each pair among them. And similarly for higher-dimensional simplices. Proteins can interact only when they are in spatial proximity, that is, when they cooccur, and so, this simplicial complex represents some kind of geometric backbone for the interaction patterns. Of course, interactions are realized by chemical affinities. This in turn leads to the question which of those potential chemical reactions are actually realized in the cell. 3D geometry may prevent certain chemically possible interactions from happening, because not all of them can happen simultaneously in space. The mathematical question then is what constraints this creates for the topology of the simplicial complex whose construction we have just described. To study such a simplicial complex, Betti numbers (dimensions of homology groups in algebraic topology, see for instance Jost (2015), ) and geometric invariants like Laplacian spectra (Horak and Jost 2013) can be used for qualitative comparison of colocalization patterns in different cells (e.g., healthy vs. diseased).
At another scale, the organization of the brain is also three-dimensional. Since not every structure can be in spatial proximity with every other structures, more distant structures need to be connected by biological wires or cables. Sending information through such cables takes time, and this then slows down the processing speed for signals entering the brain. Making the cables thicker increases the speed, but then fewer such cables can fit into some given region. Therefore, there is an optimization problem for the arrangement of the various cortical and subcortical structures and the wiring between them, so that the most important signals can be processed as fast as possible. But since those important signals and the adequate responses to them may be quite heterogeneous, compromises between the processing efficiencies of various data are necessary. Shaped by different structural constraints and channeled by historical contingencies, different brain architectures have evolved, from the distributed brains of cephalopods to the intricately folded neocortex of mammals that sits on top of and interacts with evolutionarily much older structures like the cerebellum, the basal ganglia or the hippocampus. The avian brain is much smaller and, importantly, lighter than the mammalian one and structurally differently organized, but capable of comparable intelligence. We may then ask how good the solutions are that biological evolution has found for the spatial organization of the brain, or whether another, perhaps radically different or more systematic design might be superior for the problems that the brains of current organisms have to handle.
A closed surface can shield its interior from external perturbations or influences. This inaccessibility has positive and negative aspects. An obvious positive aspect, emphasized for instance in Maturana’s and Varela’s theory of autopoiesis, is that a cell wall prevents the cell from disintegrating and at the same time, being selectively permeable enables the inflow of needed material. But then also interactions with external substances that should not or cannot be admitted into the cell need to be mediated by receptors on the cell wall and internal signaling cascades.
And we also recall that inside the cell, the DNA could not be intrinsically three-dimensional, as otherwise it would not be accessible for transcription and replication.
We conclude that information, regulation and geometric structure are interwoven, and each theoretical treatment should keep that in mind.