# Breast Cancer – How Do Scientists Apply The Cusp Bifurcation Model To Metastatic Cell State Transitions?

Bifurcation theory investigates changes in the behavior of an ordinary differential equation system when one or more parameters are altered. Concepts from cusp bifurcation theory may be used to simulate metastatic cell activity. Over time, the change of cell states may be modeled using ordinary differential equations. Bifurcation theory is a field of dynamical systems that explores how the behavior of an Ordinary differential equations system changes when one or more parameters are changed. A group of researchers headed by Brenda Delamonica and G'abor Bal'azsi from Stony Brook University in the United States demonstrated how the cusp bifurcation simulates other genetic networks and the dynamics following the bifurcation are connected to observable occurrences in commitment to join the cell cycle.

## Cusp Bifurcation And Metastatic Cell Transitions

The researchers were interested in applying mathematical models developed from bifurcation theory to explore the transitions between pro-metastatic and anti-metastatic mono-stable states of cells. The variables x(t) and parameters (t) in the model x = V (x) are time-dependent and indicate proteins involved in metastatic cell transitions. They opted to refer to the system's stable equilibria as "cell types." The cusp bifurcation is a steady two-parameter bifurcation that contains a pitchfork as a one-dimensional sub-bifurcation. They plotted the cusp curve, whicfold's the projection onto parameter space, and discovered it splits areas in the parameter space. The regions divided by the curve and the cusp point may relate to two kinds of biological transitions. The first is a continuous transition, which refers to natural phenomena in which there may be "hybrid" or "partial" cell types and happens when the dynamical system has only one stable equilibrium. This transition type may be adequate for metastasis. However, the state types and whether they constitute a continuum remain unclear. The second transition is a discontinuous transition between binary cell types or anti-metastatic and pro-metastatic cells. A stable equilibrium splits in two, making two stable equilibria and one unstable equilibrium (the saddle).

## Cusp Bifurcation In Other Networks

Aside from metastasis, the cusp point is essential in many other areas of biology as well. It should be discovered for gene networks such as bistable or multistable. This is true for the toggle switch, which is one of the fundamental gene circuits in synthetic biology. As a result, the researchers looked at 2 and 3 gene networks as instances of cusp bifurcation in biological systems. Pitchfork bifurcations have already existed in similar networks in other research.

Another substantial network drives the cell cycle. A bistable network, at the center of which is a variation of the toggle switch, guides the choice of cells between cycling and senescence. Given the analysis of the toggle switch above, the cell cycle network might alternatively be described as a toggle with a pitchfork or cusp bifurcation. Thus, the 2-gene Toggle network was used to show what may happen during cell division and how it connects to a cusp bifurcation. The toggle switch was utilized in this research to show certain physiologically relevant elements of the dynamics and to emphasize that the bifurcations of the cell cycle network will occur at various as-yet-unknown parameter values. They ascribe a biological significance to the other stable state of the toggle switch since it has two steady states, one of which corresponds to the beginning of the cell cycle. Differentiation, which is against cell cycling, is the other continuous condition.

## Conclusion

The researchers argued that the same phenomenon might be seen in 2 or 3 gene circuits and cell division dynamics that imitate "toggle" networks. In the appendix, we give a comprehensive and in-depth view of the mathematics underpinning detecting a cusp bifurcation, which we think may be extended to different biological networks. More biological systems may display cusp and pitchfork bifurcations when a simple binary transition may not adequately capture the natural occurrence. They believe it is theoretically viable to demonstrate that these systems bifurcate. If the changes in dynamics reported in this work are linked to metastatic transformations, tests must be done to see if they can be linked to them.