LE 1 Introduction to Modelling and Ordinary Differential Equations
Goals:
- Know in which cases it might be useful
- Can construct (simple) models yourself
- Can critically reflect on models in the literature
- Recognize different types of mathematical/computational models
- Translate between conceptual processes (like ‘proliferation, death, interaction’) and the
mathematical formulas and/or graphs that represent them
- Apply (simple) mathematical models to understand systems where our intuition may fail
- Analyze which assumptions are present in a given model, and assess to what extent you
agree with them
What is computational immunology?
- Data sets
- Difficult data sets analyses, modelling
- Different data putting together to solve your question
Immunoinformatics = bioinformatics for immunology
- Broadly consists of:
o Computational tools, very active field with hundred to thousands of papers per year. Many
researchers focus on developing statistical methods or data analysis frameworks.
o Databases, new databases are created at a much slower rate, in part due to stricter
standards
- Examples
o SYFPEITHI
o NetMHCPan
o The immune Epitope Database
Modelling = anything you use as a substitute for the system you are actually trying to study (for
example, because experimenting on the ‘real’ system is unethical, too costly, or too time consuming)
- Includes
o In vitro models
o In vivo models (mouse models)
o Mathematical models
o Computational models
- Two conflicting views:
o A model must be a complex as possible, otherwise it cannot faithfully represent the real
system, which is also complex
o Make things as simple as possible, but not simpler
Examples of modelling:
- Alan Turing, reaction-diffusion process, generates spots on cheetah
- Alan Perelson’s HIV model, level of plasma virus after drug therapy calculation
- Celada-Seiden model, looking at cells and there interaction and reactions
Types of models (kennen!)
- Model can be built using different formalisms:
o Equation-based models (population, numbers over time), if you describe system as one
, o Agent-based models (focus on individuals), more about details over individual units, and
differences between for example cells how they react
- And they can represent space in different ways:
o Single-compartment
o Multi-compartment
o Spatial, when you know the location
- Further models can be
o Deterministics, if you know rules at beginning then you now what outcome is (geen
kanselement)
o Stochastic, you do not know the outcome, not 100% predictable
Differential Equations describe how a certain variable changes over time
- Ordinary differential equation (ODE), dP/dt=k
- P (also called P(t)) and t are called variables and vary over time, k is a parameter. It’s value is
fixed
- Describing the change in the amount of plactic, dP/dt, rather than the current amount, P(t)
- Called differential equations because you talk about a change or growth
- Solving differential equations, means that we compute the values of our variables over time,
you need
o The differential equation itself, which describes the change over time
o The initial condition, a value for each variable at time 0
- Forward Euler Methods translate the ODE into an equation for the value at time t + 1 given
the value at time t
Exponential growth model:
- A pack of mice enters our hous. Female mice give birth about 7 times per year to about 7 pups
each time. If about half of the mice are female we can write:
o dM/dt = (0,5*M females) * (7*/yr) * (7 pups/x)
o dM/dt = 0,5 * 49 * M pups/yr
- Adjust the time resolution delta(t) of our ODE to get a better solution
- Given the time of 6 weeks to reach sexual maturity, we use delta(t) = 2months. 49 8,17 pups
every 2 months
- Mice live about 2 year, about 1/12th of our mice will die
o Modeling death dM/dt = -M/12
o dM/dt = 0,5*8,17*M – M/12
- In excel
o M(2) = =(0,5*8,17*D16)-(D16/12)+D16
o D16 = M(0) = 2 mice
Goals:
- Know in which cases it might be useful
- Can construct (simple) models yourself
- Can critically reflect on models in the literature
- Recognize different types of mathematical/computational models
- Translate between conceptual processes (like ‘proliferation, death, interaction’) and the
mathematical formulas and/or graphs that represent them
- Apply (simple) mathematical models to understand systems where our intuition may fail
- Analyze which assumptions are present in a given model, and assess to what extent you
agree with them
What is computational immunology?
- Data sets
- Difficult data sets analyses, modelling
- Different data putting together to solve your question
Immunoinformatics = bioinformatics for immunology
- Broadly consists of:
o Computational tools, very active field with hundred to thousands of papers per year. Many
researchers focus on developing statistical methods or data analysis frameworks.
o Databases, new databases are created at a much slower rate, in part due to stricter
standards
- Examples
o SYFPEITHI
o NetMHCPan
o The immune Epitope Database
Modelling = anything you use as a substitute for the system you are actually trying to study (for
example, because experimenting on the ‘real’ system is unethical, too costly, or too time consuming)
- Includes
o In vitro models
o In vivo models (mouse models)
o Mathematical models
o Computational models
- Two conflicting views:
o A model must be a complex as possible, otherwise it cannot faithfully represent the real
system, which is also complex
o Make things as simple as possible, but not simpler
Examples of modelling:
- Alan Turing, reaction-diffusion process, generates spots on cheetah
- Alan Perelson’s HIV model, level of plasma virus after drug therapy calculation
- Celada-Seiden model, looking at cells and there interaction and reactions
Types of models (kennen!)
- Model can be built using different formalisms:
o Equation-based models (population, numbers over time), if you describe system as one
, o Agent-based models (focus on individuals), more about details over individual units, and
differences between for example cells how they react
- And they can represent space in different ways:
o Single-compartment
o Multi-compartment
o Spatial, when you know the location
- Further models can be
o Deterministics, if you know rules at beginning then you now what outcome is (geen
kanselement)
o Stochastic, you do not know the outcome, not 100% predictable
Differential Equations describe how a certain variable changes over time
- Ordinary differential equation (ODE), dP/dt=k
- P (also called P(t)) and t are called variables and vary over time, k is a parameter. It’s value is
fixed
- Describing the change in the amount of plactic, dP/dt, rather than the current amount, P(t)
- Called differential equations because you talk about a change or growth
- Solving differential equations, means that we compute the values of our variables over time,
you need
o The differential equation itself, which describes the change over time
o The initial condition, a value for each variable at time 0
- Forward Euler Methods translate the ODE into an equation for the value at time t + 1 given
the value at time t
Exponential growth model:
- A pack of mice enters our hous. Female mice give birth about 7 times per year to about 7 pups
each time. If about half of the mice are female we can write:
o dM/dt = (0,5*M females) * (7*/yr) * (7 pups/x)
o dM/dt = 0,5 * 49 * M pups/yr
- Adjust the time resolution delta(t) of our ODE to get a better solution
- Given the time of 6 weeks to reach sexual maturity, we use delta(t) = 2months. 49 8,17 pups
every 2 months
- Mice live about 2 year, about 1/12th of our mice will die
o Modeling death dM/dt = -M/12
o dM/dt = 0,5*8,17*M – M/12
- In excel
o M(2) = =(0,5*8,17*D16)-(D16/12)+D16
o D16 = M(0) = 2 mice
