distributed_Biological_Systems._Chaotic_Processes.pdf

Distributed biological systems and chaotic processes in biophysics... , some models in bio and physics to explain all concepts in biophysics in this part , how to use the butterfly model and sensitive initial dependence. 

1. Distributed Biological
Systems & Chaotic
Processes
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes

2. Distributed Biological
Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes

3. Outline
Point model VS Distributed systemsPoint model VS Distributed systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Distributed system with one variable x involved in theDistributed system with one variable x involved in the
chemical process and diffused along a narrow tubechemical process and diffused along a narrow tube
ChaosChaos
Chaotic Processes in Determined SystemsChaotic Processes in Determined Systems
Models of Chaotic SystemsModels of Chaotic Systems
Model of Population DynamicsModel of Population Dynamics
Propagation of excitation waves in nervePropagation of excitation waves in nerve
and muscle tissues (Distributed system)and muscle tissues (Distributed system)
Morphogenesis as biological Distributed systemMorphogenesis as biological Distributed system

4. Point Model: focuses on analyzing variables at a specific point in space.
Example: a point model could be used to study temperature changes in
a particular city over time, without considering the spatial variations in
temperature within the city.
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Point model
Point model VS Distributed systems

5. Point model VS Distributed systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Distributed systems
These are systems composed of multiple interconnected components
that cooperate to achieve a common objective.
Example : Chemical transformations of substances can occur parallel
with the diffusion of individual substances from elementary volumes with
high concentrations to those with low concentrations, In biological
systems neighboring volumes are ( such as active membranes, tissues,
and communities of organisms)

6. Point model VS Distributed systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Distributed systems
There are also distributed sources of energy. Part of this energy is
dissipated in elementary volumes of the system. Such systems are
recognized as active distributed systems.

7. Morphogenesis as biological Distributed system
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Morphogenesis systems
It occurs spontaneously based on the information, contained in the
fertilized oocyte, in the initially spatially homogeneous medium
rather than because of external stimuli.
In this case, we have in mind the emergence of stationary three-
dimensional non-homogeneous structures in the active distributed
system.

8. Morphogenesis as biological Distributed system
Mother's Body
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Distributed resources
Uterus: (the organ in which the fetus grows).
Placenta: organ connects the growing fetus to the uterine
wall and allows for the exchange of nutrients, oxygen, and
waste products between the mother and fetus.
Umbilical Cord: connects the fetus to the placenta. It carries
oxygen and nutrients from the mother's blood to the fetus
and removes waste products.
Mechanical Forces
Cell Adhesion: Forces between cells that contribute to tissue
formation.
Cell Migration: Movement of cells within the embryo.
Apoptosis: Programmed cell death, shaping tissue and organ
formation.

9. Propagation of excitation waves in nerve and
muscle tissues (Biological distributed system)
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Multiple Interconnected Components: Nerve and muscle tissues
are composed of numerous individual cells (neurons and muscle
fibers) that interact with each other.
No Central Control: Each cell operates independently, but their
combined actions create a coordinated response.
Local Interactions: Information is exchanged between
neighboring cells, leading to a collective behavior..

10. Distributed system with one variable x involved in the chemical process
and diffused along a narrow tube
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
The model
The equation
D : is the diffusion coefficient.
I : the diffusion flux, which is the amount of
substance that diffuses per unit area per unit
time. It's a vector quantity, meaning it has both
magnitude and direction.
∂c(r,t)/∂r : concentration gradient of the
substance. It's the rate of change of
concentration with respect to position (r) at a
given time (t).
The negative sign : indicates that diffusion
occurs from regions of higher concentration to
regions of lower concentration.

11. Distributed system with one variable x involved in the chemical process
and diffused along a narrow tube
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Time-dependent change in the substance concentration : between and
depends on the difference in flows I at between and
And in the limit at
If diffusion coefficient D is constant, then
the equation for diffusion is as follows :
The time-dependent change in the substance concentration when only
diffusion proceeds in the system. And chemical reactions also take place with
“point” members

12. Distributed system with one variable x involved in the chemical process
and diffused along a narrow tube
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
If the system contains several substances :

13. Distributed system with one variable x involved in the chemical process
and diffused along a narrow tube
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Analysis of Models of Distributed Systems :
boundary conditions : if all diffusion coefficients Di = 0
at the end of the tube there may be either constant concentration of
the substance.
1.
or The ends of the tube are impermeable to diffusion flow.2.
Stationary points : should be found from the condition that time
derivatives

14. PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Stationary State: The system initially exists in a steady state represented by the
function
1.
A small disturbance : denoted as , is introduced to the system.2.
Time Evolution: The behavior of the disturbance is observed over time.3.
Stability Criteria: If the disturbance diminishes over time (t → ∞), the
system is considered stable.
4.
Factors Affecting Stability: The properties of functions f(c₁, c₂, ..., cn) influence
the system's behavior and Diffusion coefficients also play a role in determining
stability.
5.
Specific Case: In a one-dimensional system, if the (f'(c̄) < 0) , the initial deviation
will eventually disappear over time (t → ∞).
6.
Distributed system with one variable x involved in the chemical process
and diffused along a narrow tube
Analysis of Models of Distributed Systems :

15. Chaotic Processes
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes

16. Chaos refers to the complex, difficult-to-predict behavior found in
nonlinear systems.
The first recorded instance of someone noticing chaotic behavior in a
simple system was Edward Lorenz in 1960, while studying
mathematical models of weather.
One of the important properties of chaos is "sensitive dependence on
initial conditions", informally known as " the Butterfly Effect".
Sensitive dependence on initial conditions means that a very small
change in the initial state of a system can have a large effect on its later
state.
Chaos
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes

17. Deterministic Systems : These systems follow specific rules and laws . small
changes in the initial conditions lead to small alterations in the final result.
Bifurcation Points : These are critical points where small perturbations can
drastically alter a system's behavior, leading it to different possible states.
Chaotic Behavior : Despite being deterministic, many nonlinear systems exhibit
unpredictable, seemingly random behavior over time. This occurs in complex
systems with multiple variables (degrees of freedom) under specific conditions.
Origin of Chaos : This chaos arises from the system's internal dynamics, not
external influences.
Importance of Chaos : Chaotic behavior plays a crucial role in natural self-
organization processes
Chaotic Processes in Determined Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes

18. In 19-th century Poincare discovered that some mechanical
systems governed by equations of classical mechanics demonstrate
chaotic behavior.
Models of Chaotic Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
A periodically forced pendulum. Dependence of the driving force
F and angle θ versus time (according to Schuster H.G., 1984)
1.

19. Models of Chaotic Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Bernard instability (according to Schuster H.G., 1984). (a) Heat flows; (b) convective waves
generated in the liquid when temperature gradient T is higher than the critical value.
Stable Convection: At moderate temperature differences (ΔT), the fluid exhibits a
stable convective motion.
Bernard Instability: When the temperature difference (ΔT) is increased beyond a
critical point, the stable convective motion breaks down, transitioning into a chaotic
or random motion.
2. Bernard in-stability Model

20. Models of Chaotic Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
2. The Lorenz Model (1963) (deterministic)
■ Used as an atmospheric model
■ Describe the motion of a fluid layer that is heated from below in
the Bernard experiment
equations

21. x : is the fluid flow velocity ( is proportional to the velocity of the circulating
liquid)
y : is the temperature difference between the rising and falling fluid regions
z : is the difference in temperature between the top and the bottom from the
equilibrium state
r : is the control parameter proportional to the difference in temperatures.
, b : determined by the fluid properties, size of the system and the initial
temperature difference
Models of Chaotic Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
2. The Lorenz Model (1963) - variables

22. Models of Chaotic Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
2. The Lorenz Model (1963)
In this model, variables can behave chaotically at an increase in the
difference in temperatures ΔT

23. Models of Chaotic Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
2. The Lorenz Model (1963)
Phase trajectories in the Lorenz model. Top, trajectories projected onto the X-Z plane. Bottom, trajectories
projected onto the X-Y plane. Points correspond to stationary solutions (according to Lorenz, 1964).
Lorenz attractor. This is a graphical representation of the
long-term behavior of the Lorenz system of equations.
The Lorenz attractor is an example of a strange
attractor. This is a set of points in phase space towards
which a dynamical system evolves over time. Unlike
simple attractors (like fixed points or limit cycles),
strange attractors have complex structures and exhibit
sensitive dependence on initial conditions.

24. Models of Chaotic Systems
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
2. The Lorenz Model (1963) - in biological systems
The probability of chaos formation in biological systems can
be illustrated by the known example of random heart beats
at a definite frequency of stimulation pulses.

25. Model of Population Dynamics
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
The dynamics of populations in a closed environment may also have
random properties.
is dependent on the living conditions and fertility
The population dynamics can be described
discretely using a logistic equation
The population number at n+1
The population number at n
self-restricts the growth of the population number because of the
limited living space

26. If the population number is not large and at the given moment
depends on the population number in previous time periods.
The difference equation describing changes in the population
number at discrete time moments 0,1, ..., t, t+1, t+2, ... => as
corresponding numbers in sequence
Model of Population Dynamics
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Description of a logistic equation

27. The function obtained from iteration of
displays complex behavior depending on parameter r.
Model of Population Dynamics
PHYS BioPhysics : lec3-Distributed Biological Systems & Chaotic Processes
Description of a logistic equation
r is small: The system tends towards a stable state,
where the population size remains relatively
constant.
r increases: The system begins to oscillate around a
certain value, forming a regular cycle.
For higher values of r: The oscillations become
more complex, and cycles with longer periods may
appear.
At a certain critical value of r: The system's
behavior becomes chaotic, random and
unpredictable.

28. Test Bank
Give examples of chaotic systems models and explain them ?Give examples of chaotic systems models and explain them ?
PHYS BioPhysics : lec2-Enzyme kinetics
Show random properties of the dynamics of populations in a closed environmentShow random properties of the dynamics of populations in a closed environment
described by a logistic equation ?described by a logistic equation ?
What is the difference between Point model and Distributed systems ? and GiveWhat is the difference between Point model and Distributed systems ? and Give
examples on biological distributed systemsexamples on biological distributed systems
Explain Distributed system with one variable x involved in the chemical processExplain Distributed system with one variable x involved in the chemical process
and diffused along a narrow tube with equations ?and diffused along a narrow tube with equations ?

29. References
Fundamentals of Fundamentals of BiophysicsBiophysics
Andrey B. RubinAndrey B. Rubin
PHYS BioPhysics : lec2-Enzyme kinetics
https://www.slideshare.net/slideshow/chaos-https://www.slideshare.net/slideshow/chaos-
theory-48027493/48027493theory-48027493/48027493

30. Thank you
PHYS BioPhysics : lec2-Enzyme kinetics