introduction to mathematical physiology i - biochemical...

University of UtahMathematical Biology

theImagine Possibilities

Introduction to Mathematical PhysiologyI - Biochemical Reactions

J. P. Keener

Mathematics Department

University of Utah

Math Physiology – p.1/28

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 mathematics 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.2/28

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 mathematics 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.2/28

A few words about words

A big difficulty in communication between Mathematicians andBiologists is because of different vocabulary.

Examples:

• to divide -

• to differentiate -

• a PDE -

Learning the Biology Vocabulary – p.3/28

Examples:

• to divide -

• a PDE -

Examples:

• to divide - find the ratio of two numbers (Mathematician)

• a PDE -

Examples:

• to divide - replicate the contents of a cell and split into two(Biologist)

• a PDE -

Examples:

• a PDE -

Examples:

• to differentiate - find the slope of a function (Mathematician)

• a PDE -

Examples:

• to differentiate - change the function of a cell (Biologist)

• a PDE -

Examples:

• a PDE -

Examples:

• a PDE - Partial Differential Equation (Mathematician)

Examples:

• a PDE - Phosphodiesterase (Biologist)

Examples:

• a PDE - Phosphodiesterase (Biologist)

And so it goes with words like germs and fiber bundles (topologist ormicrobiologist), cells (numerical analyst or physiologist), complex(analysts or molecular biologists), domains (functional analysts orbiochemists), and rings (algebraists or protein structure chemists).

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.4/28

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

What about Galois Theory? – p.5/28

• Linear algebra

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,invasions

Places Math is Needed

• To provide quantitative theories for how biological processes work.

• Moving across spatial and temporal scales - What details toinclude or ignore, understanding emergent behaviors.

• Understanding stochasticity - Is it noise or is it real?

• The challenge of complex systems (e.g., the nature ofrobustness), to discover general principles underlying biologicalcomplexity.

• Data handling and data mining - Extracting information andfinding patterns when you don’t know what to look for, to organizeand describe the data in more comprehensible ways.

• Imaging and Visualization (Medical imaging, protein structure,etc.)

Why is Math Biology so hard? – p.7/28

• Imaging and Visualization (Medical imaging, protein structure,etc.) Why is Math Biology so hard? – p.7/28

Resources for Undergraduates

• Edelstein-Keshet (1988)

• Murray (2003)

• Segel (1984)

• Mackey & Glass (1988)

• Britton (2003)

• Ellner & Guckenheimer (2006)

• Hoppensteadt & Peskin (1992)

• Fall, Marland, Wagner, &Tyson (2002)

• Keener & Sneyd (1998)

Introduction

Biology is characterized by change. A major goal of modeling isto quantify how things change.

Fundamental Conservation Law:ddt

(stuff in Ω) = rate of transport + rate of production

In math-speak:

Ω udV =∫

∂Ω J · nds +∫

Ω fdv

production

where u is the density of the measured quantity, J is the flux of u

across the boundary of Ω, f is the production rate density, and Ω

is the domain under consideration (a cell, a room, a city, etc.)Remark: Most of the work is determining J and f !

Basic Chemical Reactions

Ak→ B

thendadt

= −ka = −dbdt

With back reactions,

thendadt

= −k+a + k−b = −dbdt

At steady state,

a = a0k−

Bimolecular Chemical Reactions

A + Ck→ B

thendadt

= −kca = −dbdt

(the "law" of mass action).

With back reactions,

A + C→

= −k+ca + k−b = −dbdt

In steady state, −k+ca + k−b = 0 and a + b = a0, so that

a = k−

k+c+k−

=Keqa0

Keq+c.

Remark: c can be viewed as controlling the amount of a.

theImagine Possibilities Example:Oxygen and Carbon

Dioxide Transport

Problem: If oxygen and carbon dioxide move into and out of theblood by diffusion, their concentrations cannot be very high (andno large organisms could exist.)

COO CO2O222

In Tissue In Lungs

CO2 O2

Problem solved: Chemical reactions that help enormously:

CO2(+H2O)→

← HCO+3 + H− Hb + 4O2

← Hb(O2)4

Hydrogen competes with oxygen for hemoglobin binding.

Dioxide Transport

−HCO3

CO2 CO2

In Tissue In Lungs

O2 −

CO2 O2

HbO 42HbO 4

CO2(+H2O)→

← HCO+3 + H− Hb + 4O2

← Hb(O2)4

Hydrogen competes with oxygen for hemoglobin binding.

Dioxide Transport

In Tissue In Lungs

CO2CO2 O

HbO 4−

CO2(+H2O)→

← HCO+3 + H− Hb + 4O2

← Hb(O2)4

Hydrogen competes with oxygen for hemoglobin binding.Why is Math Biology so hard? – p.12/28

Example II: Polymerization

monomern−mer

An + A1→

← An+1

dt= k−an+1 − k+ana1 − k−an + k+an−1a1

Question: If the total amount of monomer is fixed, what is thesteady state distribution of polymer lengths?

Remark: Regulation of polymerization and depolymerization is

fundamental to many cell processes such as cell division, cell

motility, etc.

Enzyme Kinetics

S + E→

← Ck2→ P + E

= k−c− k+sededt

= k−c− k+se + k2c = −dcdt

Use that e + c = e0, so thatdsdt

= k−(e0 − e)− k+sededt

= −k+se + (k− + k2)(e0 − e)

The QSS Approximation

Assume that the equation for e is "fast", and so in quasi-equilibrium. Then,

+ k2)(e0 − e) − k+se = 0

−+k2)e0

+k2+k+s= e0

(the qss approximation)

Furthermore, the "slow reaction" is

dt= −

= k2c = k2e0s

This is called the Michaelis-Menten reaction rate, and is used routinely (withoutchecking the underlying hypotheses).

Remark: An understanding of how to do fast-slow reductions is crucial!

Enzyme Interactions

1) Enzyme activity can be inhibited (or poisoned). For example,

S + E→

← Ck2→ P + E I + E

← C2

Then,dpdt

= −dsdt

= k2e0s

s+Km(1+ iKi

2) Enzymes can have more than one binding site, and these can"cooperate".

S + E→

← C1k2→ P + E S + C1

← C2k4→ P + E

= −dsdt

= Vmaxs2

K2m+s2

Introductory Biochemistry

• DNA, nucleotides, complementarity, codons, genes,promoters, repressors, polymerase, PCR

• mRNA, tRNA, amino acids, proteins

• ATP, ATPase, hydrolysis, phosphorylation, kinase,phosphatase

Biochemical Regulation

polymerase binding site

regulator region

"start"

Repressor bound

Polymerase bound

trpEmRNA

The Tryptophan Repressor

trpEmRNA

dt= kmP − k−mM,

dt= keM − k−eE,

dt= kRT 2R− k−RR∗, R + R∗ = R0

dt= konR∗P − koff (1− P )

dt= kT E − k−T T − 2

(Santillan & Mackey, PNAS, 2001)

Steady State Analysis

E(T ) =ke

koffR∗(T ) + 1

= k−T T,

R∗(T ) =kRT 2R0

kRT 2 + k−R

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

trp production

trp degradation

Simple example of Negative Feedback.Why is Math Biology so hard? – p.20/28

The Lac Operon

CAP binding site

RNA-polmerase binding site

-operatorstart sitelac gene

operon off(CAP not bound)

operon off(repressor bound)(CAP not bound)

operon off(repressor bound)

operon on

repressor

RNA polymerase

+ glucose

+ lactose

+ glucose

- lactose

- glucose

- lactose

- glucose

+ lactose

The Lac Operon

lac operon

repressor CAP

outside the celllactose

lactose

permease

allolactose

glucose

Lac Operon

dt= αM

1 + K1A2

K + K1A2− γMM,

dt= αP M − γP P,

dt= αBM − γBB,

dt= αLP

KLe + Le− αAB

KL + L− γLL,

dt= αAB

KL + L− βAB

KA + A− γAA.

Lac Operon - Simplified System

( P and B is qss, L instantly converted to A)

dt= αM

1 + K1A2

K + K1A2− γMM,

dt= αL

KLe + Le− βA

KA + A− γAA.

10−4 10−3 10−2 10−1 100 101 10210−4

10−3

10−2

10−1

Allolactose

dM/dt = 0

dA/dt = 0

Small Le Why is Math Biology so hard? – p.24/28

dt= αM

1 + K1A2

K + K1A2− γMM,

dt= αL

KLe + Le− βA

KA + A− γAA.

10−4 10−3 10−2 10−1 100 101 10210−4

10−3

10−2

10−1

Allolactose

dM/dt = 0

dA/dt = 0

Intermediate Le Why is Math Biology so hard? – p.24/28

dt= αM

1 + K1A2

K + K1A2− γMM,

dt= αL

KLe + Le− βA

KA + A− γAA.

10−4 10−3 10−2 10−1 100 101 10210−4

10−3

10−2

10−1

Allolactose

dM/dt = 0

dA/dt = 0

Large Le Why is Math Biology so hard? – p.24/28

Lac Operon - Bifurcation Diagram

10−3 10−2 10−1 10010−4

10−3

10−2

10−1

External Lactose

10−4 10−3 10−2 10−1 100 101 10210−4

10−3

10−2

10−1

Allolactose

dM/dt = 0

dA/dt = 0

10−4 10−3 10−2 10−1 100 101 10210−4

10−3

10−2

10−1

Allolactose

dM/dt = 0

dA/dt = 0

10−4 10−3 10−2 10−1 100 101 10210−4

10−3

10−2

10−1

Allolactose

dM/dt = 0

dA/dt = 0

Circadian Rhythms

(Tyson, Hong, Thron, and Novak, Biophys J, 1999)

Circadian Rhythms

)2 − kmM

dt= vpM −

k1P1 + 2k2P2

J + P− k3P

where q = 2/(1 +√

1 + 8KP ), P1 = qP , P2 = 12(1− q)P .

0 1 2 3 4 5 6 7 8 9 100

dM/dt = 0

dP/dt = 0

0 10 20 30 40 50 60 70 80 90 1000

time (hours)

mRNAPer

theImagine Possibilities Other Interesting Oscillatory

Networks

Glu Glu6−P

ATP ADP

F6−P F16−bisP

ATP ADP

Glycolytic Oscillations (K&S 1998)

DNA replication

Mitosis

Cyclindegradation

G1 cyclinCyclindegradation

Mitoticcyclin

Cell Cycle (K&S 1998)Why is Math Biology so hard? – p.28/28

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