Biochemical Reactions

Imagine the Possibilities

Mathematical Biology University of Utah

Introduction to Mathematical Physiology I - Biochemical Reactions

J. P. Keener

Mathematics Department University of Utah

Math Physiology – p.1/28 How can mathematics help?

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

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

Imagine the Possibilities

Mathematical Biology Introduction University of Utah

The Dilemma of Modern Biology

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

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

What can a Mathematician tell a Biologist she doesn’t already know? – p.2/28 Imagine the Possibilities

Mathematical Biology Introduction University of Utah

The Dilemma of Modern Biology

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

• The parts list is nearly complete. How the parts work together to determine function is essentially unknown. How can mathematics help?

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

• The search for emergent properties; identifying features of a collection of components that is not a feature of the individual components that make up the collection. What can a Mathematician tell a Biologist she doesn’t already know? – p.2/28 • to divide -

• to differentiate -

• a PDE -

Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

Learning the Biology Vocabulary – p.3/28 • to differentiate -

• a PDE -

Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

• to divide -

Learning the Biology Vocabulary – p.3/28 • to differentiate -

• a PDE -

Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

Learning the Biology Vocabulary – p.3/28 • to differentiate -

• a PDE -

Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

Learning the Biology Vocabulary – p.3/28 • a PDE -

Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

• to differentiate -

Learning the Biology Vocabulary – p.3/28 • a PDE -

Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

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

Learning the Biology Vocabulary – p.3/28 • a PDE -

Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

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

Learning the Biology Vocabulary – p.3/28 Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

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

• a PDE -

Learning the Biology Vocabulary – p.3/28 Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

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

• a PDE - Partial Differential Equation (Mathematician)

Learning the Biology Vocabulary – p.3/28 Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

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

• a PDE - Phosphodiesterase (Biologist)

Learning the Biology Vocabulary – p.3/28 Imagine the Possibilities

Mathematical Biology A few words about words University of Utah

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

Examples:

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

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

• a PDE - Phosphodiesterase (Biologist)

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

Learning the Biology Vocabulary – p.3/28 Space scales: Genes proteins networks cells • → → → → tissues and organs organism communities → → → ecosystems Time scales: protein conformational changes protein • → folding action potentials hormone secretion protein → → → translation cell cycle circadian rhythms human → → → disease processes population changes evolutionary → → scale adaptation

Imagine the Possibilities

Mathematical Biology Quick Overview of Biology University of Utah

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

Lots! I hope! – p.4/28 Time scales: protein conformational changes protein • → folding action potentials hormone secretion protein → → → translation cell cycle circadian rhythms human → → → disease processes population changes evolutionary → → scale adaptation

Imagine the Possibilities

Mathematical Biology Quick Overview of Biology University of Utah

Lots! I hope! – p.4/28 Imagine the Possibilities

Mathematical Biology Quick Overview of Biology University of Utah

• The study of biological processes is over many space and time scales (roughly 1016): Space scales: Genes proteins networks cells • → → → → tissues and organs organism communities → → → ecosystems Time scales: protein conformational changes protein • → folding action potentials hormone secretion protein → → → translation cell cycle circadian rhythms human → → → disease processes population changes evolutionary → → scale adaptation

Lots! I hope! – p.4/28 • 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, neural systems).

• Populations and ecosystems- biodiversity, extinction, invasions

Imagine the Possibilities

Mathematical Biology Some Biological Challenges University of Utah

• DNA -information content and information processing;

Math Physiology – p.5/28 • 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, neural systems).

• Populations and ecosystems- biodiversity, extinction, invasions

Imagine the Possibilities

Mathematical Biology Some Biological Challenges University of Utah

• DNA -information content and information processing;

• Proteins - folding, enzyme function;

Math Physiology – p.5/28 • Multicellularity - organs, tissues, organisms, morphogenesis

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

• Populations and ecosystems- biodiversity, extinction, invasions

Imagine the Possibilities

Mathematical Biology Some Biological Challenges University of Utah

• 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.?

Math Physiology – p.5/28 • Human physiology - health and medicine, drugs, physiological systems (circulation, immunology, neural systems).

• Populations and ecosystems- biodiversity, extinction, invasions

Imagine the Possibilities

Mathematical Biology Some Biological Challenges University of Utah

• 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

Math Physiology – p.5/28 • Populations and ecosystems- biodiversity, extinction, invasions

Imagine the Possibilities

Mathematical Biology Some Biological Challenges University of Utah

• 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, neural systems).

Math Physiology – p.5/28 Imagine the Possibilities

Mathematical Biology Some Biological Challenges University of Utah

• 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, neural systems).

• Populations and ecosystems- biodiversity, extinction, invasions Math Physiology – p.5/28 Imagine the Possibilities

Mathematical Biology Introduction University of Utah

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

Fundamental Conservation Law: d dt (stuﬀ in Ω) = rate of transport + rate of production In math-speak: Flux J d production dt RΩ udV = R∂Ω J nds + RΩ fdv · f(u) Ω 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! Math Physiology – p.6/28 Imagine the Possibilities

Mathematical Biology Basic Chemical Reactions University of Utah

k A B → then da = ka = db . dt − − dt With back reactions, → A B ← then da db = k a + k−b = . dt − + − dt At steady state, a = a k− . 0 k−+k+

Math Physiology – p.7/28 Imagine the Possibilities

Mathematical Biology Bimolecular Chemical Reactions University of Utah

k A + C B → then da = kca = db (the "law" of mass action). dt − − dt With back reactions, → A + C B ← da db = k ca + k−b = . dt − + − dt In steady state, k ca + k−b = 0 and a + b = a , so that − + 0 − K a a = k a0 = eq 0 . k+c+k− Keq+c Remark: c can be viewed as controlling the amount of a.

Math Physiology – p.8/28 Problem solved: Chemical reactions that help enormously: → → CO (+H O) HCO+ + H− Hb + 4O Hb(O )4 2 2 ← 3 2 ← 2 Hydrogen competes with oxygen for hemoglobin binding.

Imagine the Possibilities Example:Oxygen and Carbon

Mathematical Biology Dioxide Transport University of Utah

Problem: If oxygen and carbon dioxide move into and out of the blood by diffusion, their concentrations cannot be very high (and no large organisms could exist.)

O CO 2 CO2 O2 2

O2 CO2 O2 CO2

In Tissue In Lungs

Math Physiology – p.9/28 Hydrogen competes with oxygen for hemoglobin binding.

Imagine the Possibilities Example:Oxygen and Carbon

Mathematical Biology Dioxide Transport University of Utah

Problem: If oxygen and carbon dioxide move into and out of the blood by diffusion, their concentrations cannot be very high (and no large organisms could exist.) CO O 2 2 CO2 O2

O CO CO H − 2 2 − O 2 H 2 − − 4 4 HCO3 HCO3 HbO 2 HbO 2

In Tissue In Lungs Problem solved: Chemical reactions that help enormously: → → CO (+H O) HCO+ + H− Hb + 4O Hb(O )4 2 2 ← 3 2 ← 2

Math Physiology – p.9/28 Imagine the Possibilities Example:Oxygen and Carbon

Mathematical Biology Dioxide Transport University of Utah

Problem: If oxygen and carbon dioxide move into and out of the blood by diffusion, their concentrations cannot be very high (and no large organisms could exist.)

O CO2 2 CO2 O2

O CO2 CO 2 O 2 2 − HCO 3 4 4 HCO3 HbH HbO 2

In Tissue In Lungs Problem solved: Chemical reactions that help enormously: → → CO (+H O) HCO+ + H− Hb + 4O Hb(O )4 2 2 ← 3 2 ← 2 Hydrogen competes with oxygen for hemoglobin binding.

Math Physiology – p.9/28 Imagine the Possibilities

Mathematical Biology Example II: Polymerization University of Utah

n−mer monomer

→ An + A An 1 ← +1

dan = k−an k ana k−an + k an− a dt +1 − + 1 − + 1 1

Question: If the total amount of monomer is ﬁxed, what is the steady state distribution of polymer lengths? Remark: Regulation of polymerization and depolymerization is fundamental to many cell processes such as cell division, cell motility, etc.

Math Physiology – p.10/28 Imagine the Possibilities

Mathematical Biology Enzyme Kinetics University of Utah

→ k S + E C 2 P + E ← → ds = k−c k se dt − + de dc = k−c k se + k c = dt − + 2 − dt dp dt = k2c

Use that e + c = e0, so that ds = k−(e e) k se dt 0 − − + de = k se + (k− + k )(e e) dt − + 2 0 −

Math Physiology – p.11/28 Imagine the Possibilities

Mathematical Biology The QSS Approximation University of Utah

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

(k− + k2)(e0 − e) − k+se = 0

(k−+k2)e0 Km e = = e0 (the qss approximation) k−+k2+k+s s+Km Furthermore, the "slow reaction" is

Vmax dp − ds k c k e s dt = dt = 2 = 2 0 Km+s

s Km This is called the Michaelis-Menten reaction rate, and is used routinely (without checking the underlying hypotheses).

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

Math Physiology – p.12/28 Imagine the Possibilities

Mathematical Biology Enzyme Interactions University of Utah

1) Enzyme activity can be inhibited (or poisoned). For example, → k → S + E C 2 P + E I + E C ← → ← 2 Then, dp ds s = = k2e0 i dt dt s+Km(1+ ) − Ki 2) Enzymes can have more than one binding site, and these can "cooperate". → k → k S + E C 2 P + E S + C C 4 P + E ← 1 → 1 ← 2 →

Vmax dp ds s2 = = Vmax 2 2 dt − dt Km+s s Km

Math Physiology – p.13/28 Imagine the Possibilities

Mathematical Biology Introductory Biochemistry University of Utah

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

• mRNA, tRNA, amino acids, proteins

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

Math Physiology – p.14/28 Imagine the Possibilities

Mathematical Biology Biochemical Regulation University of Utah

polymerase binding site gene ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ¡ E

"start" regulator region ¢ ¢ ¢ ¢ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢

£ ¢ £ ¢ £ ¢ £ ¢ £ ¢ E © ¨ © ¨ © ¨ © ¨ © ¨ © ¨ © ¨ © ¨ © ¨ © ¨

¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ ¤ E

Polymerase bound

§ ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦

§ ¦ § ¦ § ¦ § ¦ § ¦ E

R* R

Of OR

DNA mRNA E trp

Math Physiology – p.15/28 Imagine the Possibilities

Mathematical Biology The Tryptophan Repressor University of Utah

R* R

Of OR

DNA mRNA E trp

dM = kmOP k−mM, dt − dOP = konOf koff OP , Of + OP + OR = 1, dt − dOR ∗ = krR Of k−rOR, dt − ∗ dR 2 ∗ ∗ = kRT R k−RR , R + R = R0 dt − dE = keM k−eE, dt − dT dR∗ = kT E k−T T 2 Math Physiology – p.16/28 dt − − dt

(Santillan & Mackey, PNAS, 2001) Imagine the Possibilities

Mathematical Biology Steady State Analysis University of Utah

ke km 1 − E(T ) = k ∗ = k T T, k−e k−m on R (T ) + 1 koff 2 ∗ kRT R0 R (T ) = 2 kRT + k−R

0.9

0.8

0.7 trp production 0.6

0.5 E(T)

0.4 trp degradation

0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T

Simple example of Negative Feedback. Math Physiology – p.17/28 Imagine the Possibilities

Mathematical Biology The Lac Operon University of Utah

CAP binding site RNA-polmerase binding site

lac gene start site -operator

+ glucose operon off + lactose (CAP not bound) repressor

+ glucose operon off (repressor bound) - lactose (CAP not bound) CAP

- glucose operon off - lactose (repressor bound) RNA polymerase

- glucose operon on + lactose

Math Physiology – p.18/28 Imagine the Possibilities

Mathematical Biology The Lac Operon University of Utah

lactose outside the cell glucose

lac permease + lac operon - + - cAMP

repressor CAP + - + lactose allolactose

k1 k2 −→ −→ R + 2A ←− RI, O + R ←− OI, k−1 k−2

E O → M → E, P, P → L −→ A

Math Physiology – p.19/28 Imagine the Possibilities

Mathematical Biology Lac Operon University of Utah

dM = αM O γM M, dt − 2 1 + K1A O = 2 (qss assumption) (-2) K + K1A dP = αP M γP P, dt − dE = αEM γEE, dt − dL Le L = αLP αAE γLL, dt KLe + Le − KL + L − dA L A = αAE βAE γAA. dt KL + L − KA + A −

Math Physiology – p.20/28 Imagine the Possibilities

Mathematical Biology Lac Operon - Simpliﬁed System University of Utah

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

2 dM 1 + K1A = αM 2 γM M, dt K + K1A − dA αP Le αE A = αL M βA M γAA. dt γP KLe + Le − γE KA + A −

2 10

1 10 dM/dt = 0

0 10

−1 10 dA/dt = 0 mRNA

−2 10

−3 10

−4 10 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 Allolactose

Small Le Math Physiology – p.21/28 Imagine the Possibilities

Mathematical Biology Lac Operon - Simpliﬁed System University of Utah

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

2 dM 1 + K1A = αM 2 γM M, dt K + K1A − dA αP Le αE A = αL M βA M γAA. dt γP KLe + Le − γE KA + A −

2 10

1 10 dM/dt = 0

0 10

dA/dt = 0 −1 10 mRNA

−2 10

−3 10

−4 10 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 Allolactose

Intermediate Le Math Physiology – p.21/28 Imagine the Possibilities

Mathematical Biology Lac Operon - Simpliﬁed System University of Utah

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

2 dM 1 + K1A = αM 2 γM M, dt K + K1A − dA αP Le αE A = αL M βA M γAA. dt γP KLe + Le − γE KA + A −

2 10

1 10 dM/dt = 0

0 10

dA/dt = 0 −1 10 mRNA

−2 10

−3 10

−4 10 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 Allolactose

Large Le Math Physiology – p.21/28 Imagine the Possibilities

Mathematical Biology Lac Operon - Bifurcation Diagram University of Utah

1 10

0 10

−1 10

−2

Allolactose 10

−3 10

−4 10 −3 −2 −1 0 10 10 10 10 External Lactose

2 2 2 10 10 10

1 1 1 10 dM/dt = 0 10 dM/dt = 0 10 dM/dt = 0

0 0 0 10 10 10

dA/dt = 0 dA/dt = 0 −1 −1 −1 10 dA/dt = 0 10 10 mRNA mRNA mRNA

−2 −2 −2 10 10 10

−3 −3 −3 10 10 10

−4 −4 −4 10 10 10 −4 −3 −2 −1 0 1 2 −4 −3 −2 −1 0 1 2 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Allolactose Allolactose Allolactose

Math Physiology – p.22/28 Imagine the Possibilities

Mathematical Biology Glycolysis University of Utah

ATP ADP − + PFK1

Glu Glu6−P F6−P F16−bisP

ATP ADP ATP ADP

k3 −→ γ γS2 + E ←− ES2 , (S2 = ADP) k−3 v 1 S , (S = ATP) −→ 1 1 k1 γ −→ γ k2 γ S1 + ES2 ←− S1ES2 ES2 + S2, k−1 −→ v S 2 . 2 −→

Math Physiology – p.23/28 Imagine the Possibilities

Mathematical Biology Glycolysis University of Utah

k3 −→ γ γS2 + E ←− ES2 k−3 v1 −→ S1 k1 γ −→ γ −k→2 γ S1 + ES2 ←− S1ES2 ES2 + S2, k−1 v2 S2 −→.

Applying the law of mass action:

ds1 = v1 − k1s1x1 + k−1x2, dt ds2 γ = k2x2 − γk3s e + γk−3x1 − v2s2, dt 2 dx1 γ = −k1s1x1 + (k−1 + k2)x2 + k3s e − k−3x1, dt 2 dx2 = k1s1x1 − (k−1 + k2)x2. dt Math Physiology – p.24/28 Imagine the Possibilities

Mathematical Biology Glycolysis University of Utah

Nondimensionalize and apply qss:

1.0

dσ1 0.8 = ν − f(σ1, σ2), dτ dσ2 0.6 = αf(σ1, σ2) − ησ2, 2

dτ σ 0.4 where γ 0.2 σ2 u1 = γ γ , σ2 σ1 + σ2 + 1 0.0 γ σ1σ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 u = 2 = f(σ , σ ). 2 γ γ 1 2 σ1 σ2 σ1 + σ2 + 1

Math Physiology – p.25/28 Imagine the Possibilities

Mathematical Biology Circadian Rhythms University of Utah

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

Math Physiology – p.26/28 Imagine the Possibilities

Mathematical Biology Circadian Rhythms University of Utah

dM vm = 2 kmM dt P2 − 1 + A dP k1P1 + 2k2P2 = vpM k P dt − J + P − 3

where q = 2/(1 + √1 + 8KP ), P = qP , P = 1 (1 q)P . 1 2 2 −

4 3.5 mRNA Per 3.5 dP/dt = 0 3

3 2.5

2.5 2

2 Per 1.5 1.5

1 1

0.5 0.5 dM/dt = 0

0 0 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 mRNA time (hours)

Math Physiology – p.27/28 Imagine the Possibilities

Mathematical Biology Cell Cycle University of Utah

Mitosis

MPF Mitotic cyclin M Cyclin degradation

G2 Cdk Cdk G1

S Cyclin Start G1 cyclin degradation SPF

DNA replication

Cell Cycle (K&S 1998)

Math Physiology – p.28/28