Chaotic Dynamical Systems and Biological Applications

Stefano Sarao

The Cell Cycle

The Protaganists

Cdk APCCyclin-dependent kineses Anaphase-promoting complex

• S-cyclins concentration remain high

• M-cyclin gene expression is switched on

• M-cyclin concentration rises

S-PhaseDna Replication

G1-PhaseLate G1 phase:

• the cell unleashes G1/S- and S-cyclin genes expression.

• G1/S-Cdk activity rises immediately

• CKIs destruction and APC inactivation

• Start

• chromosomes alignment

• spindle forming

• activation of APC

• cohesins destruction

• Cdk inhibitor production

• spindle apparatus disassembly

• cytokinesis

M-Phase

G2-Phase• S-, M-cyclins concentration remain high

• M-cyclins are activated

• G2/M Checkpoint

Concentrations

Stimulus-Response

Biological Oscillator

Biological Oscillator

Classical Chemical LawsBriefly

Michaelis-Menten equation

Goldbeter-Koshland equation

Hill equation

First application

A simple two component negative feedback system will inevitably approach a stable, intermediate state.

Tyson-Novak Modelthe cell-cycle mathematically

The Idea

formulating the problem as physicochemical process

Let’s formulate the G1/S-G2-M transition

The Idea

mass dependence (m=0.6,0.9,1.2)

The Idea

Those characterize a hysteresis loop

𝑝1 < 𝑝 < 𝑝2

Coming Back? The importance of A.

Coming Back? The importance of A.

Coming Back? The importance of A.

Coming Back? The importance of A.

Ingredients for a relaxation oscillator

Using Michaelis-Menten:

The Results

The End