EZO-23306 Modelling biological systems
Week 1 Dissecting complex systems
Introduction lecture
Biological systems are complex, mathematics can make complexity manageable, and
mathematical models make the complexity predictable. System is a limited part of reality
that contains interrelated elements.
KC 1 Modelling
A model is a simplified representation of a system. There are three types of models, physical
model, conceptual model and mathematical model. This course focuses on making
conceptual models and converting them into mathematical models. We study models to
understand, simulate and predict systems. The tools we use for this are pen and paper,
mathematics and programming(python). To solve complex models we sketch, estimate,
divide and conquer. See Appendix A for basic calculus rules.
KC 2 Back of envelop considerations
Back of envelop calculations are small calculations that do not require a calculator or
programming. Estimate the order of magnitude. Proportionality is the relationship between
two quantities in which their ratio is constant, noted as x ∝ y. Dimensions are basic concepts
of physical measurements: length, time, mass. Units are definite magnitudes of a quantity:
mm, s, kg.
KC 3 Dynamic models
dN
Biological systems change over space or time, the rate of change is denoted as , which is related
dt
dN
to the population size (over time): N ( t ). The rate of change can be per individual: r or of an
Dt
dN
population: R*N, which gives the equation: =r∗N (t ) (exponential growth).
dt
dN
If a system is in equilibrium this means no change so =0,r∗N ( t )=0 , N ( t )=0.
dt
Logistic growth means the growth rate per individual is not constant, but depends on the population
N dN N
(
size, so you assume linear decrease. r → r 1−
K )
→
dt
=r∗N (1− ), K= the carrying capacity of
K
a system.
,KC 4 introduction Jupiter notebooks
Cells in code are in python language, and cells in markdown are in text.
##text makes a heading
*text* makes text italic
**text** makes text bold
***text*** makes text bold and italic
$calculation$ for calculation in LaTeX on same line
$$calculation$$ for calculation on separate lines
$$ calculation\\calculation$$ for calculation on two lines
$$\dfrac{…}{…}=…$$ for fractions ( on points are the numbers)
$$..._...$$ is subscript
$$...^…$$ is superscript
$$\...$$ is for Greek letters (points is the exact name of Greek letter) (∝= propto)
(alpha, gamma, delta, mu, pi, rho, theta) (approx= ≈ , sqrt(…)=
√… , pm=±)
\left( ... \right)for brackets spanning more than one line
Are introduced to Jupyter notebooks and markdown cells.
Jupyter is a loose acronym of Julia, Python and R. These programming languages were the
first target languages, but nowadays the notebook also supports many other languages.
Jupyter notebook is a document that consists of code, text, equations, images etc. It
provides a complete and self-contained record of a computation or model.
, Markdown cells are used to typeset text, when run they display the corresponding rich text.
Have refreshed your mathematics skills.
See appendix A for this and toolbox exercise 1.
Know how to simplify and divide a difficult problem.
1. Divide and conquer: organise the problem into smaller problems which you can
solve.
2. Estimate: have an idea of what kind of answer you expect for the problem.
3. Sketch: sketch the situation of the problem.
Understand the concept and application of proportionality.
Proportionality is a relationship between two quantities in which their ratio is constant. This
constant is called the proportionality constant, ∝ $\propto$
Understand the concept and application of a dimensional analysis.
A dimension is a measure of a physical quantity, such as length L, time T
or mass M. A unit is a definite magnitude of a quantity, such as meters,
grams etc. A dimensional analysis is used to see if the dimension on the
left and right side of an equal sign match.(so to see if the equation is
correct)
Understand the concept and application of an order of magnitude analysis.
Making an estimate can be useful for understanding the biological system and is useful to
check your model. Order of magnitude analysis uses the symbol O. Example of the distance
from Wageningen to Rotterdam, if we would call this distance d then we can write: d=O(100
km).
Are able to identify the different components of a model.
Independent variable whose change is not influenced by other components of the
model, such as time and space
State variable that describes the state of a system, such as population size
Parameter: a constant that does not change under the influence of the state
variables or independent variables, example is rate of change.
Understand what a differential equation is and are able to analyse it graphically.
Differential equation is a mathematical equation that relates some function with its
derivatives. It describes the rate of change of a state variable as a function of the state
variable(s). In applications, the functions represent physical quantities (state variables), the
derivatives represent their rates of change, and the equation defines a relationship between
the two. You should understand from the graph that if the rate of change is positive the
state variable is increasing and if the rate of change is negative the state variable is
decreasing and if the rate of change is zero, the state variable does not change(local maxima
and minima).
Understand what an equilibrium is and what types of equilibria exist.
Week 1 Dissecting complex systems
Introduction lecture
Biological systems are complex, mathematics can make complexity manageable, and
mathematical models make the complexity predictable. System is a limited part of reality
that contains interrelated elements.
KC 1 Modelling
A model is a simplified representation of a system. There are three types of models, physical
model, conceptual model and mathematical model. This course focuses on making
conceptual models and converting them into mathematical models. We study models to
understand, simulate and predict systems. The tools we use for this are pen and paper,
mathematics and programming(python). To solve complex models we sketch, estimate,
divide and conquer. See Appendix A for basic calculus rules.
KC 2 Back of envelop considerations
Back of envelop calculations are small calculations that do not require a calculator or
programming. Estimate the order of magnitude. Proportionality is the relationship between
two quantities in which their ratio is constant, noted as x ∝ y. Dimensions are basic concepts
of physical measurements: length, time, mass. Units are definite magnitudes of a quantity:
mm, s, kg.
KC 3 Dynamic models
dN
Biological systems change over space or time, the rate of change is denoted as , which is related
dt
dN
to the population size (over time): N ( t ). The rate of change can be per individual: r or of an
Dt
dN
population: R*N, which gives the equation: =r∗N (t ) (exponential growth).
dt
dN
If a system is in equilibrium this means no change so =0,r∗N ( t )=0 , N ( t )=0.
dt
Logistic growth means the growth rate per individual is not constant, but depends on the population
N dN N
(
size, so you assume linear decrease. r → r 1−
K )
→
dt
=r∗N (1− ), K= the carrying capacity of
K
a system.
,KC 4 introduction Jupiter notebooks
Cells in code are in python language, and cells in markdown are in text.
##text makes a heading
*text* makes text italic
**text** makes text bold
***text*** makes text bold and italic
$calculation$ for calculation in LaTeX on same line
$$calculation$$ for calculation on separate lines
$$ calculation\\calculation$$ for calculation on two lines
$$\dfrac{…}{…}=…$$ for fractions ( on points are the numbers)
$$..._...$$ is subscript
$$...^…$$ is superscript
$$\...$$ is for Greek letters (points is the exact name of Greek letter) (∝= propto)
(alpha, gamma, delta, mu, pi, rho, theta) (approx= ≈ , sqrt(…)=
√… , pm=±)
\left( ... \right)for brackets spanning more than one line
Are introduced to Jupyter notebooks and markdown cells.
Jupyter is a loose acronym of Julia, Python and R. These programming languages were the
first target languages, but nowadays the notebook also supports many other languages.
Jupyter notebook is a document that consists of code, text, equations, images etc. It
provides a complete and self-contained record of a computation or model.
, Markdown cells are used to typeset text, when run they display the corresponding rich text.
Have refreshed your mathematics skills.
See appendix A for this and toolbox exercise 1.
Know how to simplify and divide a difficult problem.
1. Divide and conquer: organise the problem into smaller problems which you can
solve.
2. Estimate: have an idea of what kind of answer you expect for the problem.
3. Sketch: sketch the situation of the problem.
Understand the concept and application of proportionality.
Proportionality is a relationship between two quantities in which their ratio is constant. This
constant is called the proportionality constant, ∝ $\propto$
Understand the concept and application of a dimensional analysis.
A dimension is a measure of a physical quantity, such as length L, time T
or mass M. A unit is a definite magnitude of a quantity, such as meters,
grams etc. A dimensional analysis is used to see if the dimension on the
left and right side of an equal sign match.(so to see if the equation is
correct)
Understand the concept and application of an order of magnitude analysis.
Making an estimate can be useful for understanding the biological system and is useful to
check your model. Order of magnitude analysis uses the symbol O. Example of the distance
from Wageningen to Rotterdam, if we would call this distance d then we can write: d=O(100
km).
Are able to identify the different components of a model.
Independent variable whose change is not influenced by other components of the
model, such as time and space
State variable that describes the state of a system, such as population size
Parameter: a constant that does not change under the influence of the state
variables or independent variables, example is rate of change.
Understand what a differential equation is and are able to analyse it graphically.
Differential equation is a mathematical equation that relates some function with its
derivatives. It describes the rate of change of a state variable as a function of the state
variable(s). In applications, the functions represent physical quantities (state variables), the
derivatives represent their rates of change, and the equation defines a relationship between
the two. You should understand from the graph that if the rate of change is positive the
state variable is increasing and if the rate of change is negative the state variable is
decreasing and if the rate of change is zero, the state variable does not change(local maxima
and minima).
Understand what an equilibrium is and what types of equilibria exist.
