Introduction to Mathematical Physiology I - Biochemical Reactions

Introduction to Mathematical Physiology I - Biochemical Reactions J. P. Keener Mathematics Department Math Physiology p.1/28

Introduction 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

A few words about words A big difficulty in communication between Mathematicians and Biologists is because of different vocabulary. Examples: Learning the Biology Vocabulary p.3/28

A few words about words A big difficulty in communication between Mathematicians and Biologists is because of different vocabulary. Examples: to divide - Learning the Biology Vocabulary p.3/28

A few words about words 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

A few words about words 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 few words about words 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

Quick Overview of Biology The study of biological processes is over many space and time scales (roughly 10 16 ): Lots! I hope! p.4/28

Quick Overview of Biology The study of biological processes is over many space and time scales (roughly 10 16 ): 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

Some Biological Challenges DNA -information content and information processing; Math Physiology p.5/28

Some Biological Challenges DNA -information content and information processing; Proteins - folding, enzyme function; Math Physiology p.5/28

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

Introduction Biology is characterized by change. A major goal of modeling is to quantify how things change. Fundamental Conservation Law: d (stuff in Ω) = rate of transport + rate of production In math-speak: d Ω udv = Ω J nds + Ω fdv Flux production 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! f(u) J Math Physiology p.6/28

Basic Chemical Reactions then With back reactions, then At steady state, da A k B = ka = db. A B da = k +a + k b = db. a = a 0 k k +k +. Math Physiology p.7/28

Bimolecular Chemical Reactions then da = kca = db With back reactions, A + C k B (the "law" of mass action). A + C B da = k +ca + k b = db. In steady state, k + ca + k b = 0 and a + b = a 0, so that a = k a 0 k + c+k = K eqa 0 K eq +c. Remark: c can be viewed as controlling the amount of a. Math Physiology p.8/28

Example:Oxygen and Carbon Dioxide Transport 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 2 CO 2 O 2 CO 2 O 2 CO 2 H O CO 2 2 HbO 4 HCO 3 HbO 4 HCO 3 2 2 H In Tissue In Lungs Problem solved: Chemical reactions that help enormously: CO 2 (+H 2 O) HCO + 3 + H Hb + 4O 2 Hb(O2 ) 4 Math Physiology p.9/28

Example:Oxygen and Carbon Dioxide Transport 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 2 CO 2 O 2 CO 2 O 2 4 HbH CO 2 HCO 3 O 2 HbO 4 2 CO 2 HCO 3 In Tissue In Lungs Problem solved: Chemical reactions that help enormously: CO 2 (+H 2 O) HCO 3 + + H Hb + 4O 2 Hb(O2 ) 4 Hydrogen competes with oxygen for hemoglobin binding. Math Physiology p.9/28

Example II: Polymerization n mer monomer A n + A 1 An+1 da n = k a n+1 k + a n a 1 k a n + k + a n 1 a 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

Enzyme Kinetics S + E C k 2 P + E ds = k c k + se de = k c k + se + k 2 c = dc dp = k 2c Use that e + c = e 0, so that ds = k (e 0 e) k + se de = k +se + (k + k 2 )(e 0 e) Math Physiology p.11/28

The QSS Approximation Assume that the equation for e is "fast", and so in quasi-equilibrium. Then, or (k + k 2 )(e 0 e) k + se = 0 e = (k +k 2 )e 0 k +k 2 +k + s = e 0 K m s+k m (the qss approximation) Furthermore, the "slow reaction" is dp = ds = k 2c = k 2 e 0 s K m +s V max This is called the Michaelis-Menten reaction rate, and is used routinely (without checking the underlying hypotheses). K m s Remark: An understanding of how to do fast-slow reductions is crucial! Math Physiology p.12/28

Enzyme Interactions 1) Enzyme activity can be inhibited (or poisoned). For example, Then, S + E C k 2 P + E I + E C 2 dp = ds = k 2e 0 s s+k m (1+ i K i ) 2) Enzymes can have more than one binding site, and these can "cooperate". S + E k C 2 k 1 P + E S + C1 4 C2 P + E dp = ds = V max s 2 K 2 m +s2 V max K m s Math Physiology p.13/28

Introductory Biochemistry 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

the Imagine Biochemical Regulation R E E E E polymerase binding site regulator region gene "start" Repressor bound Polymerase bound trp E mrna DNA R* R O f O Math Physiology p.15/28

The Tryptophan Repressor R* R O f O R DNA mrna E trp dm do P do R dr de dt = k m O P k m M, = k on O f k off O P, O f + O P + O R = 1, = k r R O f k r O R, = k R T 2 R k R R, R + R = R 0 = k e M k e E, = k T E k T T 2 dr Math Physiology p.16/28

Steady State Analysis E(T ) = k e k e k m 1 k m k on k off R (T ) + 1 = k T T, R (T ) = k R T 2 R 0 k R T 2 + k R 1 0.9 0.8 0.7 0.6 trp production E(T) 0.5 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

The Lac Operon CAP binding site RNA-polmerase binding site start site -operator lac gene + glucose + lactose operon off (CAP not bound) repressor + glucose - lactose operon off (repressor bound) (CAP not bound) CAP - glucose - lactose operon off (repressor bound) RNA polymerase - glucose + lactose operon on Math Physiology p.18/28

The Lac Operon lactose outside the cell glucose lac permease + lac operon - + - camp repressor CAP + lactose + - allolactose R + 2A k 1 k 1 R I, O + R k 2 k 2 O I, O M E, P, P L E A Math Physiology p.19/28

Lac Operon dm = α M O γ M M, O = 1 + K 1A 2 K + K 1 A 2 (qss assumption) (-2) dp = α P M γ P P, de = α E M γ E E, dl L e L = α L P α A E K Le + L e K L + L γ LL, da L = α A E K L + L β A AE K A + A γ AA. Math Physiology p.20/28

Lac Operon - Simplified System ( P and B is qss, L instantly converted to A) dm da = α M 1 + K 1 A 2 K + K 1 A 2 γ MM, = α L α P γ P M L e K Le + L e β A α E γ E M A K A + A γ AA. 10 2 10 1 dm/ = 0 10 0 mrna 10 1 da/ = 0 10 2 10 3 10 4 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Allolactose Small L e Math Physiology p.21/28

Lac Operon - Bifurcation Diagram 10 1 10 0 Allolactose 10 1 10 2 10 3 10 4 10 3 10 2 10 1 10 0 External Lactose 10 2 10 2 10 2 10 1 dm/ = 0 10 1 dm/ = 0 10 1 dm/ = 0 10 0 10 0 10 0 mrna 10 1 da/ = 0 mrna 10 1 da/ = 0 mrna 10 1 da/ = 0 10 2 10 2 10 2 10 3 10 3 10 3 10 4 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Allolactose 10 4 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Allolactose 10 4 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Allolactose Math Physiology p.22/28

Glycolysis ATP PFK1 + ADP Glu Glu6 P F6 P F16 bisp ATP ADP ATP ADP γs 2 + E S 1 + ES γ 2 k 3 k 3 ES γ 2, (S 2 = ADP) v 1 S1, (S 1 = ATP) k 1 v S 2 2. k 2 S 1 ES γ 2 ES γ 2 + S 2, k 1 Math Physiology p.23/28

Glycolysis γs 2 + E k 3 k 3 ES γ 2 v 1 S 1 S 1 + ES γ 2 k 1 k 2 S 1 ES γ 2 ES γ 2 + S 2, k 1 v S 2 2. Applying the law of mass action: ds 1 ds 2 dx 1 dx 2 = v 1 k 1 s 1 x 1 + k 1 x 2, = k 2 x 2 γk 3 s γ 2 e + γk 3x 1 v 2 s 2, = k 1 s 1 x 1 + (k 1 + k 2 )x 2 + k 3 s γ 2 e k 3x 1, = k 1 s 1 x 1 (k 1 + k 2 )x 2. Math Physiology p.24/28

Glycolysis Nondimensionalize and apply qss: 1.0 dσ 1 dτ = ν f(σ 1, σ 2 ), 0.8 where dσ 2 dτ = αf(σ 1, σ 2 ) ησ 2, σ 2 0.6 0.4 u 1 = σ γ 2 σ γ 2 σ 1 + σ γ 2 + 1, 0.2 0.0 u 2 = σ 1 σ γ 2 σ γ 2 σ 1 + σ γ 2 + 1 = f(σ 1, σ 2 ). 0.0 0.2 0.4 0.6 σ 1 0.8 1.0 1.2 Math Physiology p.25/28

Circadian Rhythms (Tyson, Hong, Thron, and Novak, Biophys J, 1999) Math Physiology p.26/28

Circadian Rhythms dm dp = v m 1 + ( P 2 A ) 2 k m M = v p M k 1P 1 + 2k 2 P 2 J + P k 3 P where q = 2/(1 + 1 + 8KP ), P 1 = qp, P 2 = 1 2 (1 q)p. 4 3.5 dp/ = 0 3.5 3 mrna Per 3 2.5 2.5 2 Per 2 1.5 1.5 1 1 0.5 dm/ = 0 0.5 0 0 1 2 3 4 5 6 7 8 9 10 mrna 0 0 10 20 30 40 50 60 70 80 90 100 time (hours) Math Physiology p.27/28

Cell Cycle Mitosis Mitotic cyclin MPF M Cyclin degradation Cdk G 2 G 1 Cdk Cyclin degradation S Start G 1 cyclin SPF DNA replication Cell Cycle (K&S 1998) Math Physiology p.28/28