Mathematics is biology's next microscope,

only better;

Biology is mathematics' next physics,

only better.

Joel E Cohen

Here, I wrote about a few of these mathematical lensings I have already worked on in several opportunities.

**Networks** (or graphs) are mathematical tools we use to model real problems (in biology, sociology, geography, economy,…) and study them systematically. In particular, it is possible to study *expanding networks*, given a node addition rule. We studied numerically the growth of some of these networks based on averages of random initialization and analytically, through a **master equation formulation** applied to these addition rules, accounting for all possible scenarios that may probabilistically happen within the model.

The importance of this work is predicting the future behavior of the real systems described by the studied networks and understanding and quantifying the parameters related to this modeling. Here, we consider **Divergence and Duplication** networks, which mimic properties of *Protein-Protein Interaction Networks (PPI)* - i.e., given **mutation parameters**, the characteristic features of these graphs are similar to those found on experimental networks of related proteins.

We established the parameter ranges where the mean number of connections between nodes converges to finite number, and when it diverges making the network more and more connected as we add new nodes. **When the total mutation exceeds a threshold, the mean degree of D&D networks diverge as the networks grow.** Check the visual distinctions where divergence is increased from left to right.

This work is a collaboration with professors **Leonardo G. Brunnet**, Rita M. C. de Almeida, Ricardo M. Ferreira and Daniel Gamermann within LabCel group at the Physics Institute at UFRGS. It was funded by federal government (CNPq). All results of this project are published on the following article at *Physica A*.

Master equation for the degree distribution of a Duplication and Divergence network | |||

Sudbrack, V.; Brunnet, L.G.; de Almeida, R.M.; Ferreira, R. M. & Gamermann, D. | |||

Physica A: Statistical Mechanics and its Applications↗ |
2018 |

You can also check the presentation below with a synthesis of the main results: determining parameter ranges for which networks reach a steady state and evaluating the time-dependent and asympthotic network degree distribution.

**Magnetosensibility** is the ability of organisms to respond to the presence of a magnetic field, changing their behavior. This can be achieved actively through **magnetoreception**, when organisms detect the geomagnetic field and use this vector information in their spatial orientation, or passively, as in **magnetotaxia**, when just like a needle, organisms behave as magnetic dipoles and align themselves with the lines of the Earth’s magnetic field. These kinds of responses have been observed, for instance, in dogs, ants, several birds and bees.

In order to further understand this kind of behaviour in a **Multicellular Magnetotactic Prokaryotes (MMP)** collected in Brazilian lake, we designed this experimental work aiming to study the **kinetic description** of a particular type of movement (*ping-pong motion* or **escape mobility**) in MMPs as the intensity of the magnetic field varies, as well as its probability of occurrence. Its characteristic of being **stratified in one dimension** can be seen in the following video-shot, where the left plot of parallel position varies in order of magnitude more than the perpendicular position (the right plot).

Our main conclusion was that this type of motility has several characteristics that depend on the magnetic field intensity and, therefore, it is best explained by **magnetoreceptive mechanisms**. All results are publish in the following article from *European Biophysics Journal*.

Magnetoreception in multicellular magnetotactic prokaryotes: a new analysis of escape motility trajectories in different magnetic fields | |||

Sepulchro, A.G.V.; de Barros H.L.; de Mota H.O.L.; Berbereia K.S.; Huamani, K.P.T.; Lopes, L.C. da S.; Sudbrack, V. & Acosta‐Avalos, D. | |||

European Biophysics Journal↗ |
2020 |

This work happened during the 4th Advanced School of Experimental Physics (**4th EAFExp**) organized by **Centro Brasileiro de Pesquisas Físicas (CBPF)** in February 2019, supervised by professors **Daniel Acosta-Avalos** and Henrique Lins de Barros, and done together with some amazing young researchers I had the chance to meet through this project: Ana Gabriela, Henrique, Karen, Katterine and Lis.

You can also check a summary of main points in this group presentation at the school (in Brazilian portuguese).