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5.1. HAPLOID GENETICS
65
genotype
freq. of gamete
relative tness
freq after selection
freq after mutation
normalization
A
a
p
q
1
1 s
p/w
(1 s)q/w
(1u)p/w [(1 s)q + up]/w
w = p + (1 s)q
Table 5.3: A haploid genetic model of mutationselection balance.
ca
The Hong Kong University of Science and Technology
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Spring 2014
6.4. ALLOSTERIC INHIBITION
k1 S
E
6 k
k
k3 I
3
k0
k2
C1
 P +E
6
1
k0
0
k3 I
0
k1 S
?
C2

97
 ?
C3
3
0
k2
 P + C2
1
The general model for allosteric inhibition with ten independent rate constants appears too complicated to analyze. We will simplify thi
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Mathematical Biology
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Spring 2014
7.2. BRUTE FORCE ALIGNMENT
105
Figure 7.2: James Watson and Francis Crick posing in front of their DNA
model. The original photograph was taken in 1953, the year of discovery, and
was recreated in 2003, fty years later. Francis Crick, the man on the right
The Hong Kong University of Science and Technology
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Spring 2014
5.2. DIPLOID GENETICS
73
genotype
freq. of zygote
relative tness
AA
Aa
aa
p2
2pq
q2
1
1 sh
1 s
p2 /w 2(1 sh)pq/w (1 s)q 2 /w
w = p2 + 2(1 sh)pq + (1 s)q 2
freq after selection
normalization
Table 5.9: A diploid genetic model of mutationselection balance
The Hong Kong University of Science and Technology
Mathematical Biology
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Spring 2014
Chapter 6
Biochemical Reactions
Biochemistry is the study of the chemistry of life. It can be considered a branch
of molecular biology, perhaps more focused on specic molecules and their reactions, or a branch of chemistry focused on the complex chemical
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Mathematical Biology
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Spring 2014
4.4. VACCINATION
Disease
Diphtheria
Haemophilus
inuenzae type
b (Hib)
Hepatitis A
Description
A bacterial respiratory disease
A
bacterial
infection occurring primarily in
infants
A viral liver disease
Hepatitis B
Same as Hepatitis A
Measles
A viral respir
The Hong Kong University of Science and Technology
Mathematical Biology
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Spring 2014
2.7. THE BROOD SIZE OF A HERMAPHRODITIC WORM
33
only parameter that depends on Band apply the condition dr/dB = 0. We
nd
(2.23)
(r + d) exp (g + B/p)(r + d) = p exp (B/m)(r + d) .
Taking the ratio of (2.22) to (2.23) results in
mn
1=
exp (B/m)(r + d)
p
fr
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Spring 2014
3.3. SIMULATION OF POPULATION GROWTH
49
1
y = F() = 1 ebN
y2=F(2)
y1=F(1)
0
1
2
Figure 3.1: How to generate a random number with a given probability density
function.
R
numbers in (1 , 2 ) = 12 P ( )d . This is exactly the denition of having
probability d
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Chapter 1
Population Dynamics
Populations grow in size when the birth rate exceeds the death rate. Thomas
Malthus, in An Essay on the Principle of Population (1798), used unchecked
population growth to famously predict a global famine unless governments r
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2.5. DISCRETE AGESTRUCTURED POPULATIONS
25
Here, we have made the simplifying assumption that n
! so that all the
females counted in the n + 1 census were born after the rst census.
The highorder linear dierence equation (2.8) may be solved using the
an
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1.4. THE LOTKAVOLTERRA PREDATORPREY MODEL
9
terms model contact between predators and prey. The parameter is the fraction of prey caught per predator per unit time; the total number of prey caught
by predators during time t is
tU V . The prey eaten is t
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3.2. ASYMPTOTICS OF LARGE INITIAL POPULATIONS
3.2
41
Asymptotics of large initial populations
Our goal here is to solve for an expansion of the distribution in powers of 1/N0
to leadingorders; notice that 1/N0 is small if N0 is large. To zerothorder, th
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2.2. THE GOLDEN RATIO
2.2
17
The golden ratio
The number
is known as the golden ratio. Two positive numbers x and y,
with x > y, are said to be in the golden ratio if the ratio between the sum of
those numbers and the larger one is the same as the ratio b