Introduction to Mathematical Physiology I - Biochemical Reactions
J. P. Keener
Mathematics Department University of Utah
Math Physiology-p. 1/28
Mathematical Biology University of Utah
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.
What can a Mathematician tell a Biologist she doesn't already know? - p.2/28
Mathematical Biology University of Utah
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
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rA 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
University of Utah
rA 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
University of Utah
rA 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
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
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
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
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
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
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
A few words about words
University of Utah
rA 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
Mathematical Biology University of Utah
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
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Quick Overview of Biology
The study of biological processes is over many space and time scales (roughly 1016):
Lots! I hope! -p.4/28
Quick Overview of Biology
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• 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
Lots! I hope! -p.4/28
Mathematical Biology University of Utah
Quick Overview of Biology
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 scale adaptation
population changes —► evolutionary
Lots! I hope! -p.4/28
Some Biological Challenges
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• DNA -information content and information processing;
Math Physiology - p.5/28
Mathematical Biology University of Utah
Some Biological Challenges
DNA -information content and information processing Proteins - folding, enzyme function;
Math Physiology - p.5/28
Mathematical Biology University of Utah
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.?
Math Physiology - p.5/28
Mathematical Biology University of Utah
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.?
Multicellular^ - organs, tissues, organisms, morphogenesis
Math Physiology - p.5/28
Mathematical Biology University of Utah
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.?
Multicellular^ - organs, tissues, organisms, morphogenesis
Human physiology - health and medicine, drugs, physiological systems (circulation, immunology, neural systems).
Math Physiology - p.5/28
Mathematical Biology University of Utah
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.?
Multicellular^ - 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
Mathematical Biology University of Utah
Introduction
Biology is characterized by change. A major goal of modeling is to quantify how things change.
Fundamental Conservation Law:
^ (stuff in fi)    rate of transport + rate of production
In math-speak:
Flux
Í In udv = fan J ' nds + In fdv
where u is the density of the measured quantity, J is the flux of u across the boundary of fi, / is the production rate density, and Vt is the domain under consideration (a cell, a room, a city, etc.) Remark: Most of the work is determining J and /!
Math Physiology - p.6/28
Imagine the
Ǥ
Mathematical Biology I University of Utah
da dt
Basic Chemical Reactions
A^B
da _ _7 V7 _ _db
dt ~     ^a ~ dt'
A^B
= -k+a + k-b =
k-
Math Physiology - p.7/28
University of Utah
k
A + C A B
then
da dt
= —kca =
With back reactions.
^   (the "law" of mass action)
A + C^B
da
dt =-k+ca + k_b=-ft.
In steady state, -k+ca + k-b = 0 and a + b = a0, so that
a =
_ k-ao _ Keqao
Remark: c can be viewed as controlling the amount of a.
Math Physiology - p.8/28
Mathematical Biology University of Utah
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.)
o2 CO2 O2 CO.
In Tissue
In Lungs
Math Physiology - p.9/28
Mathematical Biology University of Utah
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.)
co2 02
In Tissue In Lungs
Problem solved: Chemical reactions that help enormously:
C02{+H20) ^ HCO+ + H~      Hb + 402 ^ Hb{02)A
Math Physiology - p.9/28
Mathematical Biology University of Utah
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, C02 02 co2
CO,
----	
	
In Lungs
In Tissue
Problem solved: Chemical reactions that help enormously:
C02{+H20) ^ HCO+ + H~ Hb + 402 ^ Hb{02)A Hydrogen competes with oxygen for hemoglobin binding.
Math Physiology - p.9/28
Mathematical Biology University of Utah
Example II: Polymerization
n-mer
monomer
An + A
A
n+l
da
n
dt
= /c_an+i — /c+anai — k-an + /c+an_iai
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
Mathematical Biology University of Utah
Enzyme Kinetics
S + E
cH P + E
ft=k-c
k+se
%= k-c - k+se + k2c = -I
dl = k2C
Use that e + c = e0, so that
§ = /c_(e0 - e) - fc+se § = -k+se + (k- + fc2)(e0 - e)
_
Math Physiology - p. 11/28
Possibilities
Mathematical Biology University of Utah
The QSS Approximation
Assume that the equation for e is "fast", and so in quasi-equilibrium. Then,
(k- + fe)(eo — e) — k+se = 0
or
e =
(fc_+fc2)e0 fc_ -\-k2Jrkjrs
= e0
K,
__m
s+Kr,
(the qss approximation)
Furthermore, the "slow reaction" is
dp _
dt ~
-ft=k2c = k2e0l^T-s
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
Mathematical Biology University of Utah
Enzyme Interactions
1) Enzyme activity can be inhibited (or poisoned). For example.
S + E^chp + E      I + E^C2
Then
% = -|=^eo
S+Km(l+-^)
2) Enzymes can have more than one binding site, and these can "cooperate".
S + E^d^P + E      S + C1^C2k^P + E
dp _ ds
dt
_ _ OS _ TT"
- J J- - V 7
dt
max _l.s2
Math Physiology - p. 13/28
Mathematical Biology University of Utah
Introductory Biochemistry
DNA, nucleotides, complementarity, codons, genes, promoters, repressors, polymerase, PCR
imRNA, tRNA, amino acids, proteins
ATP, ATPase, hydrolysis, phosphorylation, kinase, phosphatase
Math Physiology - p. 14/28
Biochemical Regulation
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polymerase binding site
gene
"start"
regulator region
E
E
E
Math Physiology - p. 15/28
The Tryptophan Repressor
R*
R
It
DNA
mRNA
*- E-► trp
km Op k—rnM.
konOf - k0ffOp,      0f + 0p + 0R = 1, krR*Of — k-rOR, kRT2R - k-RR\       R + R* = R0 keM - k-eE,
kTE - k-pT - 2
dR
dt
Math Physiology - p. 16/28
Mathematical Biology University of Utah
Steady State Analysis
E(T) =
R* (T) =
J\ip k
m
k—& k—
k
m
on
koff
R*(T) + 1
= k-TT,
kRT2Ro kRT* + k.R
Possibilities
Mathematical Biology University of Utah
The Lac Operon
CAP binding site
/ RNA-polmerase binding site
i-1
start site — -operator
lac gene
+ glucose + lactose
+ glucose - lactose
glucose lactose
- glucose + lactose
		LAAAAAAAAAKCWWc		—
				
CAP
repressor
RNA polymerase
operon off (CAP not bound)
operon off (repressor bound) (CAP not bound)
operon off (repressor bound)
operon on
Math Physiology - p. 18/28
The Lac Operon
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lactose
outside the cell
glucose
+
^repressorj    ^ CAP ...
lactose-
allolactose
kl k2
R + 2A^Ri,      O + R^Oi.
-l
-2
E
O^m^e,P, P^L^a
Math Physiology - p. 19/28
Lac Operon
dM ~dt
1 + KxA2
O   =   —-——777     (qss assumption) (-2)
K + KiA2     v '
oipM — ^)pP} aEM - jEE,
dP
~dt ~ dE
~dt ~
dL Le L
dt KLe + Le KL + L
dA „    L        n „ A
-dl   = aAEK^TL-ßAEKlTÄ-lAA
Math Physiology - p.20/28
Mathematical Biology University of Utah
Lac Operon - Simplified System
( P and B is qss, L instantly converted to A)
dM ~dT dA ~dt
1 + ^A2
aMK + KlAi-7MM'
op „, aL—M
L,
jp    KLe + Le        je    KA + A
7A A.
/   dM/dt = 0
i / i ' i ' i >
t /
dA/dt = 0 ,'/'
10 10 10
Allolactose
Small L,
Math Physiology - p.21/28
Mathematical Biology University of Utah
Lac Operon - Simplified System
( P and B is qss, L instantly converted to A)
dM ~dT dA ~dt
1 + ^A2
aMK + KlAi-7MM'
op „, aL—M
L,
jp    KLe + Le        je    KA + A
7A A.
y'   dM/dt = 0
dA/dt = 0
10 10 10
Allolactose
Intermediate Lt
Math Physiology - p.21/28
Mathematical Biology University of Utah
Lac Operon - Simplified System
( P and B is qss, L instantly converted to A)
dM ~dT dA ~dT
1 + ^A2
aMK + KlAi-7MM'
op „, aL—M
L,
jp    KLe + Le        je    KA + A
7A A.
10 10 10
Allolactose
Large L(
Math Physiology - p.21/28
Lac Operon - Bifurcation Diagram
External Lactose
Math Physiology - p.22/28
Glycolysis
ATPV ADP
PFK1
+
Glu
ATP ADP
Glu6-P
F6-P
ATP ADP
F16-bisP
7S2 + E
k<
k_3 Vi
ESI
Si,
(S2 = ADP) (Si = ATP)
7
k-
Si + ES2'
S-
V2
SiES^ES] + S2
Math Physiology - p.23/28
Imagine the
Possibilities
Mathematical Biology University of Utah
Glycolysis
7S2 +E
Si + ES2
S2
Applying the law of mass action:
dsi
fe_3 ->
V2
ES]
S-
SiES?^ES^ + S2.
dt
ds 2
dt
dx
—    k2X2 — ^k3s^e + 7^-3x1 — t>2s2 ■
_
da: 2
dt
-    =    — k\s\x\ + (k-i + ^2)^2 + fcßs^e — ksxi, =    kisixi — (k-i + k2)x2-
Math Physiology - p.24/28
Mathematical Biology University of Utah
Glycolysis
Nondimensionalize and apply qss:
da i dr
d(T2
dr
where
U\ =
U2 =
=      V - /(č7l,č72),
=    af (0-1,0-2) - W2
o
7
er £ er i + ö\J + 1
7
o2o\ + o2 + 1
1.0 -1 0.8 -0.6 -0.4-0.2 -0.0 -
—i-1-1-1-1-r
0.0     0.2     0.4     0.6     0.8     1.0 1.2
°i
Math Physiology - p.25/28
Mathematical Biology University of Utah
Rhythms
AAAAAA
Nucleus
i xxxxxxxxxxxx
(tím)
AAAAAA
Cytoplasm
(Tyson, Hong, Thron, and Novak, Biophys J, 1999)
Math Physiology - p.26/28
Mathematical Biology University of Utah
Circadian Rhythms
dM ~dT
dP ~dt
v
m
kjYi Is/L
vpM -
\ A !
fciPi + 2k2P2
J + P
-hP
where q = 2/(1 + y/l + 8KP), Pi = qP, P2 = i(l - q)P.
dM/dt = 0
0 1 2 3 4 5 6 7
mRNA
90 100
Math Physiology - p.27/28
Cell Cycle
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Mitosis
Mitotic cyclin
Cyclin degradation
Cyclin degradation
G-| cyclin
DNA replication
Cell Cycle (k&s 1998)
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