(dynamic_properties_of_biological_process)bio_physics.pdf

Dynamic Properties of
Biological Processes
PHYS LECL:1_BIOPHYSICS Eng/ Mahana M Elbana - @KFS university - mahanaelbana@gmail.com

Outline
6 Reduction of the Number of Equations
1
3
2
4
Biological Kinetics
Elementary Model of Open System
The Feedback Principle
Qualitative Analysis of the Model
5 Stationary Point Stability
PHYS LECL:1_BIOPHYSICS Eng/Mahana M Elbana

Biological Kinetics
Biological kinetics is a branch of biology that studies the interactions
between cells and their environment.
These interactions include the exchange of materials between cells and
the environment, as well as the internal metabolic processes that take
place within cells.
Biological kinetics consists of a complex network of chemical and physical
interactions that occur between the different components of cells. These
components include biomolecules such as [proteins, enzymes, sugars, and
DNA]. These components interact with each other in a sequential and
interconnected manner, which leads to the emergence of the different
biological processes that characterize living organisms.

Biological Kinetics
Factors affecting biological kinetics:
The surrounding environment: such as temperature, humidity, and pH
affect the rate of chemical and physical reactions that occur within cells.
Nutrients: Cells need nutrients such as carbohydrates, proteins, and fats
to obtain the energy they need.

Biological Kinetics
Analyzing a Closed Cell Population ( multiplication and death occur
concurrently with abundant in nutrients):
Aim to :
How does the number of cells change over time?
1.
Can a stable state (stationary state) be reached where the cell population
remains constant?
2.
assumption:
Let at moment (t) the concentration of cells in the environment be N.
The rate of the cell concentration changes dN/dt in the environment is
the net sum of their multiplication rate and death rate.

Biological Kinetics
the net sum of their multiplication rate and death rat
PHYS LECL:1_BIOPHYSICS
k1 is the proportionality constant dependent on the environmental
conditions (temperature, the presence of nutrients).
k2 is the constant determining the intensity of processes of cell
death
where k = k1 -k2 ,By solving the equation
we will see how the cell concentration is
changed with time in the environment N
= N(t) ,By integrating.
N0 is the cell concentration at zero
time t = 0 of the examining the system. Eng/Mahana M Elbana

Biological Kinetics
Notes :
If k1 > k2, k > 0, the system will give rise to the unlimited
growth of the cell number.
If k2 < k1 , the population will dye out with time
when k1 = k2 the number of cells will remain constant
Eng/Mahana M Elbana

Biological Kinetics
The Verhulst equation
the model of the population growth in the environment with a limited
amount of nutrients is the known equation of a logistic curve
N(max) is the maximal population number
possible under such conditions.
At the initial period of growth, when N << N(max)
the curve is exponential (blue curve)
After the inflection, the slope gradually decreases
and the curve approaches the upper asymptote
N = N(max), i.e. the maximal attainable level under
such conditions (red curve The Verhulst equation).
Eng/Mahana M Elbana

Spatial
Heterogeneity
Variables
Beyond
Concentration
Spatial
Dependence
(Diffusion)
Biological Kinetics vs Chemical Kinetics
Key
Differences
1. Variables Beyond Concentration:
Chemical kinetics primarily focuses on concentration
of reactants as the main variable.
Biological kinetics incorporates additional variables
like:
Presence of other molecules: Activators and
inhibitors can significantly influence reaction rates.
Cellular environment: Factors like pH and
temperature play a role in biological processes.
Biological Kinetics
Self-
Regulation
(Feedback)
Non-Integer
Reaction
Orders
2. Spatial Dependence (Diffusion):
Chemical kinetics often assumes a well-mixed
system where reactants readily collide.
Biological systems often involve spatial
compartments separated by membranes.
Diffusion of molecules across these membranes
becomes a crucial factor in reaction rates.
Eng/Mahana M Elbana

Spatial
Heterogeneity
Variables
Beyond
Concentration
Spatial
Dependence
(Diffusion)
Key
Differences
3. Spatial Heterogeneity:
Chemical kinetics often assumes a uniform reaction
environment.
Biological systems are heterogeneous, meaning
conditions and concentrations can vary throughout a
cell.
This can lead to localized variations in reaction
rates.
Biological Kinetics
Self-
Regulation
(Feedback)
Non-Integer
Reaction
Orders
4. Self-Regulation (Feedback):
Chemical reactions typically lack inherent control
mechanisms.
Biological systems often exhibit self-regulation
through feedback loops.
Products of a reaction can influence its own
rate (positive or negative feedback).
This allows for more precise control of cellular
processes. Eng/Mahana M Elbana

Spatial
Heterogeneity
Variables
Beyond
Concentration
Spatial
Dependence
(Diffusion)
Key
Differences
5. Non-Integer Reaction Orders:
Chemical kinetics often relies on integer reaction
orders based on stoichiometry.
Biological reactions can involve complex enzyme
mechanisms and multiple steps.
This can lead to non-integer reaction orders
in the differential equations describing them.
Biological Kinetics
Self-
Regulation
(Feedback)
Non-Integer
Reaction
Orders
Eng/Mahana M Elbana

Feedback in The Hydrodynamic Model
Hydrodynamic model with a special device that can increase or decrease the
rate of liquid outflow upon rotation of the faucet at the outlet of the vessel
depending on the liquid level changes.
Eng/Mahana M Elbana

Feedback in The Hydrodynamic Model
model component:
A reservoir with a faucet at the outlet.
A light source and photocell that sense the water level.
An electromotor that controls the faucet opening based on the photocell
signal.
A small turbine powered by the water outflow that supplies electricity to the
lamp and motor.
aim to:
In this model the feedback principle maintains to a certain extent the liquid
level upon varying the water inflow as a result of self-regulation
Eng/Mahana M Elbana

Feedback in Biological Systems
The importance of feedback of biological systems:
regulating many enzyme reactions where the activity of
enzymes varies depending on the reagent concentration
or external conditions.
Eng/Mahana M Elbana

Analyzing an elementary model of an open system which exchanges
substances a and b with the environment
the model refl ects the basic features of metabolic processes in a cell
For example, glucose and oxygen as substrates for respiration are
supplied to a cell at stage
corresponds to the release of
CO2 and Н2O to the outside of the
cell, and the entire metabolic
respiratory cycle of the glucose
molecule transformation can be
described by reaction
Eng/Mahana M Elbana

The kinetic equations for this system are as follows:
At a stationary state, variables (а, b) are constant, then
Eng/Mahana M Elbana

Let us equate the right-hand side of eq. (1.4) to zero:
We get the system of algebraic equations:
Eng/Mahana M Elbana

Not depend on the initial conditions
depend only on the constant values and substance concentrations in the
external vessels A and B.
The system of
differential
equations (1.4) is
solvable if
dependences a = a(t)
and b = b(t)
Eng/Mahana M Elbana

Not depend on the initial conditions

show the system attains a stationary state independent of the initial conditions

Figure 1.4 demonstrates several types of such transition curves a(t). Similar
shapes of the curves were traced, for example, in physiological studies of the
respiration rate under different initial conditions
values of а and b, they determine a specific shape of the curves of changes in a(t)
and b(t) and the kinetics of the transition of the system from the starting point:
The shape of the curves depends on the initial conditions and constant
Explanation of the curve
1
2
3
equations (1.4) have only linear members in their right-hand sides with unknown variables
to the first power. However as a rule in biological systems, many processes are non-linear
Eng/Mahana M Elbana

Qualitative Behavior, Not Exact Solutions:
Finding exact solutions to the complex differential equations used in these
models can be challenging. Instead, qualitative analysis aims to understand
the general trends and characteristics of the system's behavior.
Qualitativeanalysis focuses on:
Types of Steady States: Identifying both stable (system returns after
disruptions) and unstable steady states.
Transitions Between States: Understanding how the system moves from one
steady state to another.
Oscillatory Regimes: Exploring cases where the system fluctuates around a
specific value.
Sensitivity to Parameters: Analyzing how changes in key parameters within
the model affect the system's behavior.
Eng/Mahana M Elbana

Revealing General Properties: This approach helps uncover essential
characteristics of the model without needing to explicitly solve for unknown
functions.
Benefits of Qualitative Analysis:
Stability: A Key Feature:
The passage emphasizes the importance of "stability" in a steady state. A
stable system can return to its original state after external disturbances.
Eng/Mahana M Elbana

Stationary Point Stability
The kinetic equation is :
k is the rate constant v(outfl)
The simplest open system (substance a is supplied from an external source at
a constant rate vo = v(infl) = const).
At stationary point:
Seek always , the concentration at
the stationary point or vo
Eng/Mahana M Elbana

decrease in stationary concentration of
Stationary Point Stability
Disturbance occurred in the stationary-state system
The simplest open system (substance a is supplied from an external source at
a constant rate vo = v(infl) = const).
raising the stationary concentration of
1
2
Accidental deviations from the stationary point are compensated by
the system itself which just means that the system is in a stable stationary.
3
the system should decrease the inflow rate until the
rates (vinfl = voutfl) ==> use
the outflow rate would grow until vo =v(outfl)
Eng/Mahana M Elbana

Reduction of the Number of Equations
Reduction in the number of equations cannot be made arbitrarily and hence
accomplishment should obey the objective laws and rules. Otherwise it is
hazardous to lose some important properties that will make the model
inadequate to the simulated biological system.
Conditions of reduction
Stability: A Key Feature:
Analyzing Right-Hand Sides: Studying the properties of the right-hand sides of
differential equations, even without exact solutions, can reveal essential features
of stable states within a system.
Need for Reduction: This is why biophysicists seek methods to reduce the number
of equations in the initial model. The goal is to obtain a simpler model with fewer
equations that still captures the crucial dynamic properties of the system.
Eng/Mahana M Elbana

Test Bank
Q1: Give two examples on Biological Kinetics Cell Population, one of them using
abundant in nutrients, the other using limited amount of nutrients(a logistic
curve) , show your answer using equations and figures?
Q2: mention Key Differences between Biological Kinetics and Chemical Kinetics?
Q4: Stationary Point Stability is one of Dynamic Properties of Biological
Processes .Give an example of it and the cases of deviation from the Stationary
Point. Explain your answer using equations?
Q5: mention key features and conditions of Reduction of the Number of
Equations ?
Q3: mention the importance of feedback principles in biological systems?
Eng/Mahana M Elbana

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