mathematical modelling for biosciences - exeter.ac.uk · today session motivation for mathematical...
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Mathematical Modelling for Biosciences
Daniel GalvisKyle Wedgwood10th July 2019
Centre for Biomedical Modelling and Analysis
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Course content
Motivation/development of mathematic models
Simulation of models
Phase space analysis and predictions
Write/understand code/equations
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Today session
Motivation for mathematical modelling
The mathematics of modelling
Chemical Reactions
Neural Models
Population Models-
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What is a mathematical model?
Mathematical Model: A mathematical representation of real-world systems. We will focus on ”rates of changes” of things such as populations, chemical concentrations, etc.
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How are they developed?
Bottom-up models – building up complex systems from simple experimental results (gene pathway models/HH model)
Top-down models - finding the simplest, necessary parts to model a higher level system (predator-prey models)
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How are they developed?
Mechanistic – more complex, try to “stay true” the the underlying biological processes that lead to the behaviour of interest
Phenomenological - simple models that try to capture a behaviour, generally in the simplest way possible
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Extracellular space
Intracellular space
CICa IK IL
RCa RK RL
VCa VK VL
Hodgkin-Huxley Model Fitzhugh-Nagumo Model
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Why do we use them?• Integrate many pieces of information
• Quick and cheap to simulate (compared to lab work)
• Make Predictions
• Bridge Scales
• Simplification (mechanistic -> phenomenological)
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Limitations• Curse of dimensionality
• Parameter uncertainty
• Computational cost
• Noise
• Models can be wrong
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The modelling paradigm
“All models are wrongbut some are useful”
- George Box
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The modelling paradigm
• A faithful representation of the real world system
• Able to be simulated/analysed
• Constructed at an appropriate scale
• Able to provide testable predictions
• Flexible and adaptable
• Simple (but not too simple)
A good model should be
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The modelling paradigm
“Everything should be made as simple as possible
but no simpler”
- Albert Einstein
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The mathematics
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Xppaut• Software for simulating and analysing mathematical models
• Written by Bard Ermentroutwww.math.pitt.edu/~bard/xpp/xpp.html
• Essentially a graphical front end for a library of numerical routines to simulate equations
• Reads in text files encoding models
• Easy to use with either mouse or keyboard
• Can be used with python
• Now available on iPad!
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Predator prey model• Simple phenomenological model of one predator species and
one prey species
• Overly simplistic, since it ignores all other organisms, but gives general overview of predatory dynamics
• Has also found use in neural modelling, opinion modelling and game theory
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Lotka-Volterra model• Originally a model of chemical reactions, but was shortly
afterwards applied to population modelling
• Let denote the amount of grass and denote the number of rabbits
• Note that these are both continuous (rather than discrete variables). This makes sense if the populations of both are big enough
• In the model grass grows exponentially quickly with rate and rabbits die exponentially quickly with rate
• Grass gets eaten by the rabbits at a rate , and the rabbits breed at a rate, , dependent on the availability of grass
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Lotka-Volterra equations• Grass grows:
• Rabbits eat grass:
• Rabbits breed after eating:
• Rabbits die:
• Full model
• Behaviour dependent on parameters and initial conditions
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Simulating the Lotka-Volterra model
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Simulating the Lotka-Volterra model
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The Brusselator
Belousov-ZhabotinskyReaction
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The Brusselator• 6 chemical species
• Well mixed environment
• Constant supply of , immediate clearing of
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Law of Mass Action• Simple model of chemical reactions
• Rate of reaction is directly proportional to the product of the masses of the reactants
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Our first model
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Neural modelling• Essentially, almost all biophysical models of neurons are based
on the 1952 formalism by Hodgkin and Huxley
• Originally developed to describe the behaviour of the squid giant axon
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Hodgkin-Huxley model
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Hodgkin-Huxley model
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Extracellular space
Intracellular space
CINa IK IL
RNa RK RL
VNa VK VL
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Morris-Lecar model• Variant of the Hodgkin-Huxley model which replaces sodium
current with a calcium current
• Hodgkin-Huxley model has 4 state variables, Morris-Lecar has only 2
• Originally a model of:
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Morris-Lecar model
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Extracellular space
Intracellular space
CICa IK IL
RCa RK RL
VCa VK VL
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Morris-Lecar model
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Morris-Lecar model
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Extracellular space
Intracellular space
CICa IK IL
RCa RK RL
VCa VK VL
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Calcium current
conductance Nernst potential
prop. open channels
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Full model
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Extra Slides
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Neural networks
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Drug delivery
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Endocrinology
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Cancer