Waves and vortices in nature

Biophysics: Biophysics is the study of physical phenomena and physical processes in living things, on scales spanning molecules, cells, tissues and organisms. Biophysicists use the principles and methods of physics to understand biological systems.

https://www.youtube.com/watch?time_continue=142&v=yMY72jLHoK8

Metabolic pathway

Metabolic pathway

Protein kinase A pathway

As a result: in complex biological systems change on one parameter results in changes of many other variables

Huge variability: Body consists of cells of many different types

->Even two adjacent cells of the same organ can differ their properties more than 30%

->Connexin43 (Cx43) gap junction protein has a half-life of about 1–3 h, other channels also in the course of hours

-> Ionic channels change their properties depending on the phase of the day

From P. Kohl

In this course we will study
an approach for multiscale modelling
based on Reaction-Diffusion equations

In this course we will study mostly waves in the heart
However, waves are present in many biological systems.
We will study trigger waves: waves which switch a system from from one state to another state

Example 1:

state 0: grass

state 1: burned grass

Example 2: boiling of water with a wire

state 0: boiling from water

state 1: boiling from gas

= Heat transfer

state 0

state 1

transition

In grass: after growing of the grass, state 1 can go back to state 0 = recovery time of the system

Time it takes for the system to go back to state 0

These are excitable media! Example of the grass is an excitable medium

Example 3: mexican wave

Excitable media:

Have a resting state (pre-excitation)
Excitation
Refractory period

Most important example: the heart is an excitable medium

property 1: colliding waves

colliding waves annihilate

property 2: two periodic sources

when having two periodic sources, the fastest source will win

property 3: formation of spiral waves

[L] = m

[R] = s

[v] = m/s

property 3: formation of spiral waves

Example 4: Belousov-Zhabotinsky reaction

The mechanism for this reaction is very complex and is thought to involve around 18 different steps which have been the subject of a number of research papers.

Most simple representation

Zhabotinsky has shown that the oscillations in the solution color were due to oscillations in concentration of Ce4+

The reaction works as follows:

Suppose we have a large concentration of Ce 4+ : then Br − is produced rapidly, which inhibits Ce 3+ → Ce 4+
thus the Ce 4+ concentration drops and as a result Br − concentration drops.
Because of this the autocatalytic reaction Ce 3+ → Ce 4+ starts again, the Ce 4+ concentration increases,
The cycle repeats itself.

The ‘fuel’ of this reaction is the malonic acid (reducer) and the oxidizer (bromate). They are, however, gradually consumed in this reaction, preserving the law of conservation.

The main idea behind these oscillations in the BZ reaction is just the classical idea of a negative feedback mechanism, which can be explained in the following informal qualitative way.

Assume that we set fire to a match and that it starts burning. The fire will produce a lot of smoke, and the smoke will stop/reduce the fire as it needs oxygen to burn.
Without the oxygen, the fire will reduce and the smoke will be cleared by the wind, resulting in a new burning flare.
This however, is not in contradiction with the law of conservation of energy: each oscillation consumes some energy (each time the match becomes shorter) and finally the oscillations will stop.

CO reacts with oxygen on the Pt surface, forms CO2
From these two molecules O2 binds to the Pt surface stronger than CO.
If the surface is covered by CO, it prevents O2 from binding to the surface (because CO is larger than O2 ). If the surface is covered by O2 , CO can still bind. Obviously this can lead to bistability, where state one is the surface completely covered by the oxygen and state two is the surface is completely covered by CO (poisoned catalyser).
The main regime here is the propagation of the CO state.

Example 5:

Nobel Prize Chemistry in 2007

Example 6: Ca waves

Ca is very toxic for the cells
Normally, cell tries to keep its internal concentration of Ca low
Ca is stored in a special compartment called the sarcoplasmatic reticulum
Cardiac contraction requires Ca
Autocatalytic process: Ca-induced-Ca-release: increase of Ca concentration close to the SR induces Ca-release etc
Ca will diffuse as a wave through the cell

Example 6: Ca waves

Ca waves are also present during the fertilization of the egg by a sperm-cell. Here a Ca wave will propagate along the membrane
Sometimes the propagation of a circular wave is distorted and results in the formation of spatiotemporal pictures

Example 7: Dictyostelium discoidum

The slime Dictyostelium discoidum is a species of amoeba

Individual cells can communicate using cyclic-AMP

c-AMP is produced by cells in response to stress
c-AMP acts as a chemo-attractant for other cells
c-AMP makes cells produce more c-AMP: so you get a kind of activation
c-AMP production becomes refractory

There is a positive feedback and refractoriness, which suggests that a group of Dictyoatelium cells can als act as an excitable medium

How does this cAMP cause otherwise free-living cells to form a "super-organism"?

To understand this, a 3D spatial model of Dictyostelium discoidum was created.

This was modeled by describing the processes as a differential equation

Problem:

If there’s a source of cAMP, the amoebae moves toward the source and collect at this location.

Problem

However, as we already know, the fastest source inducing the waves will win in an excitable environment. Hence, the aggregation of the amoebae will start to form at the place were the the amoeba with the fastest period is located (period = the pace at with an amoeba can resend the cAMP signal).
This mechanism solely depends on the single fastest amoebae. So that if this amoeba dies or stops firing fast due to a change in environmental conditions, the complete amoebae-system will be in chaos as all amoebae which were under the way to that fastest amoeba will not know where to go anymore.

This can be solved by forming spiral waves: