Mathematical Models in Biology

Postgraduate Thesis uoadl:1446810 941 Read counter

Unit:
Κατεύθυνση Εφαρμοσμένα Μαθηματικά
Library of the School of Science
Deposit date:
2017-03-01
Year:
2017
Author:
Vasilopoulou Eleftheria
Supervisors info:
Ιωάννης Στρατής, Καθηγητής, Μαθηματικών, ΕΚΠΑ
Λεώνη Ευαγγελάτου- Δάλλα, Καθηγήτρια, Μαθηματικών, ΕΚΠΑ
Γεράσιμος Μπαρμπάτης, Αναπληρωτής καθηγητής, Μαθηματικών, ΕΚΠΑ
Original Title:
Μαθηματικά Μοντέλα στη Βιολογία
Languages:
Greek
Translated title:
Mathematical Models in Biology
Summary:
The increasing amount of biological experimental data in all branches of
biology, in conjunction with the possibilities offered by mathematics, promotes
the creation of many prediction models for biological systems. The research
usually begins by running experiments on a biological subject and by tackling the
complexity of subcellular data in order to proceed to modeling and the
understanding of factors that need to be considered or ignored so that the certain
model will be successful. The experience of applications in biology shows that,
through this cooperation there is as much progress in one science, as there is in the
other and in the last century this interaction lead to a rapid development.
At all levels of biological societies, i.e. in molecular, cells, organisms,
populations, communities and ecosystems, the modeling of biological process is
particularly important. Of course the concept “model” in biology is open to many
interpretations. Some ideas are described with qualitative models; other models
are formed mathematically such as those that described prey – predator
interactions with differential equations. There are even more complex models
constructed to capture more subtle elements from many real data systems, some of
which constitute a system of partial differential equations. Biological systems
present specific difficulties in being solved because of the large numbers of types
of objects and of the objects of every type but also because of the complexity of
the interaction between objects.
Success has been noted at all levels (molecular, cells, organisms,
populations, communities and ecosystems). In this master thesis models are
presented, along with the relevant mathematical techniques that have proven to be
productive in biology. The concept of model of ordinary differential equations is
introduced, their formation, analysis and interpretation, through a simple, as to its
description, experiment in microbiology on the development of unicellular
microorganisms. In particular, the model regards the bacteria and the change in
their population, at given time, in relation to the nutrients consumed, while the
experiments are performed in laboratory conditions, in a medium called
chemostat. The model has bases on Malthus’ allegations about the impact of
population growth but also on Verhulst’s logistic model. Given the fact that the
chemostat model is complex in its interpretation, it seems that the only solutions
that can be found analytically are stable steady states. The stability properties of
these solutions are of major importance, as in biological problems there are
always perturbations and the question is whether deviations from steady state lead
to big changes or not. Examples are presented on mathematical description of
chemostat with significant clinical applications, as in cancer, where ultimately the
results of chemotherapy, that is the response of the cell undergoing treatment, is
determined by dynamic interaction of several biological agents, such as the
orientation of treatment in cells, mechanisms of the drug action, development and
differentiation of cell populations and resistance.
In many cases, equations that describe a biological problem are not linear,
so a useful method for the analysis of stability is the linearization which, however,
can provide local information. Models that aim at the description of phenomena
where interactions between atoms, spaces and population lead to relations that
depend on variables in a complex manner, contain nonlinear equations which are
difficult, if not impossible, to be solved in a closed and analytical way. However,
the calculation of such solutions is not always necessary since the qualitative
elements can be determined geometrically in conjunction with basic geometric
ideas and intuition. The geometric theory provides the ability for this as it makes
the recovery of an image that provides understanding of the way the parameters
and constants that appear in equations affect the behavior of the system, is
achieved.
The geometric theory is applied in this study on the interpretations of
solutions for the chemostat. It is then applied to population models on which, if
there is an ecosystem in question, two or more types of interactions and therefore
the population of one of them, cannot be studied independently from the
population of the other types. The Lotka-Volterra models of prey – predator are
presented, as well as models of competition, symbiosis and the epidemiological
model SIR of Kermack and McKendrick.
In addition to the above, mathematic models are widely used to describe
chemical reactions contemplated by the chemical kinetic and their parameters,
such the speed with which they are carried out, the experimental conditions
affecting the speed, the reaction mechanisms, but also the transition state. A
method with which we can easily form differential equations that arise from
chemical reactions, is presented and mathematically, the conservation laws as a
result of fundamental equations in natural sciences. Subsequently the present
master thesis is extended to the enzymatic reactions, the differential equations that
describe them and phenomena of inhibition, cooperativity and quasi steady state
methods are studied along with the Michaelis-Menten model. The presented
methods include the method of singular perturbations which enable approximate
solutions of the problem, in cases in which the equations that describe involve
small terms.
Cooperativity has a central role in multistable systems, in storage and
development, where hyperbolic and sigmoidal responses arise. The cell
differentiation is studied, through Lewis’s model, in which bifurcations have a key
role. Furthermore, gives the fact periodicity is a phenomenon inherent in living
things and the periodic behavior appears in cell division, in the signal emitted by
neurons etc, the study of the stability of periodical phenomena becomes important.
Concepts of periodic orbits and limit cycles are introduced. The Poincare –
Bendixson theorem is introduced as well as basic ideas on the bifurcation theory
which is used when the parameters of a system slightly change. Finally, the
Hodgkin and Huxley model (one of the most significant models) is presented: it
describes the potential energy in cells and is an example of a particularly
successful interaction between Biology and Mathematics.
Main subject category:
Science
Other subject categories:
Mathematics
Keywords:
Mathematical models
Apply in biology
Geometric theory
Index:
No
Number of index pages:
0
Contains images:
Yes
Number of references:
22
Number of pages:
150
File:
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