University of UtahMathematical Biology

theImagine Possibilities

A Brief Introduction to Mathematical BiologyChallenges and Opportunities

J. P. Keener

Mathematics Department

University of Utah

Math Biology: Challenges and Opportunities – p.1/7

University of UtahMathematical Biology

theImagine Possibilities

Introduction

The Dilemma of Modern Biology

• The amount of data being collected is staggering. Knowingwhat to do with the data is in its infancy.

• The parts list is nearly complete. How the parts worktogether to determine function is essentially unknown.

How can mathematicians help?

• The search for general principles; organizing and describingthe data in more comprehensible ways.

• The search for emergent properties; identifying features of acollection of components that is not a feature of theindividual components that make up the collection.

What can a Mathematician tell a Biologist she doesn’t already know? – p.2/7

University of UtahMathematical Biology

theImagine Possibilities

Introduction

The Dilemma of Modern Biology

• The amount of data being collected is staggering. Knowingwhat to do with the data is in its infancy.

• The parts list is nearly complete. How the parts worktogether to determine function is essentially unknown.

How can mathematicians help?

• The search for general principles; organizing and describingthe data in more comprehensible ways.

• The search for emergent properties; identifying features of acollection of components that is not a feature of theindividual components that make up the collection.

What can a Mathematician tell a Biologist she doesn’t already know? – p.2/7

University of UtahMathematical Biology

theImagine Possibilities

Quick Overview of Biology

• The study of biological processes is over many space andtime scales (roughly 1016):

• Space scales: Genes → proteins → networks→ cells→tissues and organs → organism → communities →

ecosystems

• Time scales: protein conformational changes → proteinfolding → action potentials → hormone secretion → proteintranslation → cell cycle → circadian rhythms → humandisease processes → population changes → evolutionaryscale adaptation

Lots! I hope! – p.3/7

University of UtahMathematical Biology

theImagine Possibilities

Quick Overview of Biology

• The study of biological processes is over many space andtime scales (roughly 1016):

• Space scales: Genes → proteins → networks→ cells→tissues and organs → organism → communities →

ecosystems

• Time scales: protein conformational changes → proteinfolding → action potentials → hormone secretion → proteintranslation → cell cycle → circadian rhythms → humandisease processes → population changes → evolutionaryscale adaptation

Lots! I hope! – p.3/7

University of UtahMathematical Biology

theImagine Possibilities

Quick Overview of Biology

• The study of biological processes is over many space andtime scales (roughly 1016):

• Space scales: Genes → proteins → networks→ cells→tissues and organs → organism → communities →

ecosystems

• Time scales: protein conformational changes → proteinfolding → action potentials → hormone secretion → proteintranslation → cell cycle → circadian rhythms → humandisease processes → population changes → evolutionaryscale adaptation

Lots! I hope! – p.3/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around.

What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around.

What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around.

What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around.

What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around.

What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around.

What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around.

What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Kinds of Math I have Used

• Discrete Math - graph theory, finite state automata,combinatorics

• Linear algebra

• Probability and stochastic processes

• ODE’s, PDE’s, Delay DE’s, Integro-Differential Equations, ...

• Numerical Analysis

• Topology - knots and scrolls, topological invariants

• Algebraic Geometry, Projective geometry

Editorial Comment: Let the biology dictate the math, not the other

way around. What about Galois Theory? – p.4/7

University of UtahMathematical Biology

theImagine Possibilities

Some Biological Challenges

• DNA -information content and information processing;

• Proteins - folding, enzyme function;

• Cell - How do cells move, contract, excrete, reproduce,signal, make decisions, regulate energy consumption,differentiate, etc.?

• Multicellularity - organs, tissues, organisms, morphogenesis

• Human physiology - health and medicine, drugs,physiological systems (circulation, immunology, neuralsystems).

• Populations and ecosystems- biodiversity, extinction,invasion

Math Biology: Challenges and Opportunities – p.5/7

University of UtahMathematical Biology

theImagine Possibilities

Some Biological Challenges

• DNA -information content and information processing;

• Proteins - folding, enzyme function;

• Cell - How do cells move, contract, excrete, reproduce,signal, make decisions, regulate energy consumption,differentiate, etc.?

• Multicellularity - organs, tissues, organisms, morphogenesis

• Human physiology - health and medicine, drugs,physiological systems (circulation, immunology, neuralsystems).

• Populations and ecosystems- biodiversity, extinction,invasion

Math Biology: Challenges and Opportunities – p.5/7

University of UtahMathematical Biology

theImagine Possibilities

Some Biological Challenges

• DNA -information content and information processing;

• Proteins - folding, enzyme function;

• Cell - How do cells move, contract, excrete, reproduce,signal, make decisions, regulate energy consumption,differentiate, etc.?

• Multicellularity - organs, tissues, organisms, morphogenesis

• Human physiology - health and medicine, drugs,physiological systems (circulation, immunology, neuralsystems).

• Populations and ecosystems- biodiversity, extinction,invasion

Math Biology: Challenges and Opportunities – p.5/7

University of UtahMathematical Biology

theImagine Possibilities

Some Biological Challenges

• DNA -information content and information processing;

• Proteins - folding, enzyme function;

• Cell - How do cells move, contract, excrete, reproduce,signal, make decisions, regulate energy consumption,differentiate, etc.?

• Multicellularity - organs, tissues, organisms, morphogenesis

• Human physiology - health and medicine, drugs,physiological systems (circulation, immunology, neuralsystems).

• Populations and ecosystems- biodiversity, extinction,invasion

Math Biology: Challenges and Opportunities – p.5/7

University of UtahMathematical Biology

theImagine Possibilities

Some Biological Challenges

• DNA -information content and information processing;

• Proteins - folding, enzyme function;

• Cell - How do cells move, contract, excrete, reproduce,signal, make decisions, regulate energy consumption,differentiate, etc.?

• Multicellularity - organs, tissues, organisms, morphogenesis

• Human physiology - health and medicine, drugs,physiological systems (circulation, immunology, neuralsystems).

• Populations and ecosystems- biodiversity, extinction,invasion

Math Biology: Challenges and Opportunities – p.5/7

University of UtahMathematical Biology

theImagine Possibilities

Some Biological Challenges

• DNA -information content and information processing;

• Proteins - folding, enzyme function;

• Cell - How do cells move, contract, excrete, reproduce,signal, make decisions, regulate energy consumption,differentiate, etc.?

• Multicellularity - organs, tissues, organisms, morphogenesis

• Human physiology - health and medicine, drugs,physiological systems (circulation, immunology, neuralsystems).

• Populations and ecosystems- biodiversity, extinction,invasion

Math Biology: Challenges and Opportunities – p.5/7

University of UtahMathematical Biology

theImagine Possibilities

Places Math is Needed

• Moving across spatial and temporal scales - What details toinclude or ignore, understanding emergent behaviors (Thereis no such thing as a homogeneous environment ormedium).

• Understanding stochasticity - Is it noise or is it real?(Periodic behaviors do not exist in biology)

• The challenge of complex systems (e.g., the nature ofrobustness).

• Data handling and data mining - Extracting information andfinding patterns when you don’t know what your are lookingfor.

• Imaging and Visualization (Medical imaging, proteinstructure, etc.)

Math Biology: Challenges and Opportunities – p.6/7

University of UtahMathematical Biology

theImagine Possibilities

Places Math is Needed

• Moving across spatial and temporal scales - What details toinclude or ignore, understanding emergent behaviors (Thereis no such thing as a homogeneous environment ormedium).

• Understanding stochasticity - Is it noise or is it real?(Periodic behaviors do not exist in biology)

• The challenge of complex systems (e.g., the nature ofrobustness).

• Data handling and data mining - Extracting information andfinding patterns when you don’t know what your are lookingfor.

• Imaging and Visualization (Medical imaging, proteinstructure, etc.)

Math Biology: Challenges and Opportunities – p.6/7

University of UtahMathematical Biology

theImagine Possibilities

Places Math is Needed

• Moving across spatial and temporal scales - What details toinclude or ignore, understanding emergent behaviors (Thereis no such thing as a homogeneous environment ormedium).

• Understanding stochasticity - Is it noise or is it real?(Periodic behaviors do not exist in biology)

• The challenge of complex systems (e.g., the nature ofrobustness).

• Data handling and data mining - Extracting information andfinding patterns when you don’t know what your are lookingfor.

• Imaging and Visualization (Medical imaging, proteinstructure, etc.)

Math Biology: Challenges and Opportunities – p.6/7

University of UtahMathematical Biology

theImagine Possibilities

Places Math is Needed

• Moving across spatial and temporal scales - What details toinclude or ignore, understanding emergent behaviors (Thereis no such thing as a homogeneous environment ormedium).

• Understanding stochasticity - Is it noise or is it real?(Periodic behaviors do not exist in biology)

• The challenge of complex systems (e.g., the nature ofrobustness).

• Data handling and data mining - Extracting information andfinding patterns when you don’t know what your are lookingfor.

• Imaging and Visualization (Medical imaging, proteinstructure, etc.)

Math Biology: Challenges and Opportunities – p.6/7

University of UtahMathematical Biology

theImagine Possibilities

Places Math is Needed

• Moving across spatial and temporal scales - What details toinclude or ignore, understanding emergent behaviors (Thereis no such thing as a homogeneous environment ormedium).

• Understanding stochasticity - Is it noise or is it real?(Periodic behaviors do not exist in biology)

• The challenge of complex systems (e.g., the nature ofrobustness).

• Data handling and data mining - Extracting information andfinding patterns when you don’t know what your are lookingfor.

• Imaging and Visualization (Medical imaging, proteinstructure, etc.)

Math Biology: Challenges and Opportunities – p.6/7

University of UtahMathematical Biology

theImagine Possibilities

Summary

• What you will learn here is barely the tip of the iceberg.

• The challenges are enormous, but so are the opportunities.

• Explore, Experiment, Create, Work Hard, Have Fun!

Math Biology: Challenges and Opportunities – p.7/7

University of UtahMathematical Biology

theImagine Possibilities

Summary

• What you will learn here is barely the tip of the iceberg.

• The challenges are enormous, but so are the opportunities.

• Explore, Experiment, Create, Work Hard, Have Fun!

Math Biology: Challenges and Opportunities – p.7/7

University of UtahMathematical Biology

theImagine Possibilities

Summary

• What you will learn here is barely the tip of the iceberg.

• The challenges are enormous, but so are the opportunities.

• Explore, Experiment, Create, Work Hard, Have Fun!

Math Biology: Challenges and Opportunities – p.7/7