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Mathematical Modelling for Biosciences Daniel Galvis Kyle Wedgwood 10th July 2019 Centre for Biomedical Modelling and Analysis

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Page 1: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Mathematical Modelling for Biosciences

Daniel GalvisKyle Wedgwood10th July 2019

Centre for Biomedical Modelling and Analysis

Page 2: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Course content

Motivation/development of mathematic models

Simulation of models

Phase space analysis and predictions

Write/understand code/equations

Page 3: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Today session

Motivation for mathematical modelling

The mathematics of modelling

Chemical Reactions

Neural Models

Population Models-

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Page 4: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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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Page 5: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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)

Page 6: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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

Page 7: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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)

Page 8: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Limitations• Curse of dimensionality

• Parameter uncertainty

• Computational cost

• Noise

• Models can be wrong

Page 9: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

The modelling paradigm

“All models are wrongbut some are useful”

- George Box

Page 10: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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

Page 11: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

The modelling paradigm

“Everything should be made as simple as possible

but no simpler”

- Albert Einstein

Page 12: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

The mathematics

Page 13: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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!

Page 14: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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

Page 15: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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

Page 16: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Lotka-Volterra equations• Grass grows:

• Rabbits eat grass:

• Rabbits breed after eating:

• Rabbits die:

• Full model

• Behaviour dependent on parameters and initial conditions

Page 17: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Simulating the Lotka-Volterra model

Page 18: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Simulating the Lotka-Volterra model

Page 19: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

The Brusselator

Belousov-ZhabotinskyReaction

Page 20: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

The Brusselator• 6 chemical species

• Well mixed environment

• Constant supply of , immediate clearing of

Page 21: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Law of Mass Action• Simple model of chemical reactions

• Rate of reaction is directly proportional to the product of the masses of the reactants

Page 22: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Our first model

Page 23: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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

Page 24: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Hodgkin-Huxley model

Page 25: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Hodgkin-Huxley model

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Extracellular space

Intracellular space

CINa IK IL

RNa RK RL

VNa VK VL

Page 26: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

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:

Page 27: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Morris-Lecar model

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Extracellular space

Intracellular space

CICa IK IL

RCa RK RL

VCa VK VL

Page 28: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Morris-Lecar model

Page 29: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Morris-Lecar model

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Extracellular space

Intracellular space

CICa IK IL

RCa RK RL

VCa VK VL

Page 30: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Calcium current

conductance Nernst potential

prop. open channels

Page 31: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Full model

Page 32: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Extra Slides

Page 33: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Neural networks

Page 34: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Drug delivery

Page 35: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Endocrinology

Page 36: Mathematical Modelling for Biosciences - exeter.ac.uk · Today session Motivation for mathematical modelling The mathematics of modelling Chemical Reactions Neural Models Population

Cancer