6.2.1 For
the given probability that a molecule leaves a cell, write the
discrete-time dynamical system for the probability that it remains
inside (assuming it can never return) and find the solution. Compute the
probability that the molecule remains inside after 10 seconds, and the
time before it will have left with probability 0.9. The probability it leaves is 0.3 each second.
Get solution
6.2.2 For
the given probability that a molecule leaves a cell, write the
discrete-time dynamical system for the probability that it remains
inside (assuming it can never return) and find the solution. Compute the
probability that the molecule remains inside after 10 seconds, and the
time before it will have left with probability 0.9. The probability it leaves is 0.03 each second.
Get solution
6.2.3 The
following probabilities describe molecules that can hop into and out of
a cell. For each, find a discrete-time dynamical system for the
probability that the molecule is inside. Find the probability that a
molecule that begins inside is inside at t =2, and the probability that a molecule that begins outside is inside at t =2.
Compute the equilibrium, and use it to estimate how many out of 100
molecules would be inside after a long time. The probability it leaves
is 0.3 each second, and the probability it returns is 0.2 each second.
Get solution
6.2.4 The
following probabilities describe molecules that can hop into and out of
a cell. For each, find a discrete-time dynamical system for the
probability that the molecule is inside. Find the probability that a
molecule that begins inside is inside at t =2, and the probability that a molecule that begins outside is inside at t =2.
Compute the equilibrium, and use it to estimate how many out of 100
molecules would be inside after a long time. The probability it leaves
is 0.03 each second, and the probability it returns is 0.1 each second.
Get solution
6.2.5
Draw cobweb diagrams based on the discrete-time dynamical systems in
the earlier exercise. The molecule in Exercise 1. Reference Exercise 1
The probability it leaves is 0.3 each second.
Get solution
6.2.6
Draw cobweb diagrams based on the discrete-time dynamical systems in
the earlier exercise. The molecule in Exercise 2. Reference Exercise 2
The probability it leaves is 0.03 each second.
Get solution
6.2.7
Draw cobweb diagrams based on the discrete-time dynamical systems in
the earlier exercise. The molecule in Exercise 3. Reference Exercise 3
The probability it leaves is 0.3 each second, and the probability it
returns is 0.2 each second.
Get solution
6.2.8
Draw cobweb diagrams based on the discrete-time dynamical systems in
the earlier exercise. The molecule in Exercise 4. Reference Exercise 4
The probability it leaves is 0.03 each second, and the probability it
returns is 0.1 each second.
Get solution
6.2.9 In many ways, probabilities act like fluids. For each of the following models of chemical exchange, let ...represent the amount in container 1 and ...the amount in container 2 at time t. Write a discrete-time dynamical system for the amount of chemical in each container. Define ... to be the fraction of chemical in container 1, and write a discrete-time dynamical system giving ...as a function of ... . Find the equilibrium fraction of chemical in the first container. Each
second, 30% of the chemical in container 1 enters container 2, and 20%
of the chemical in container 2 returns to container 1 (compare with
Exercise 3).
Get solution
6.2.10 Each
second, 3% of the chemical in container 1 enters container 2, and 10%
of the chemical in container 2 returns to container 1 (compare with
Exercise 4).
Get solution
6.2.11
Compute the following probabilities for a selfing plant using Figure
6.2.9. The fraction of grand-offspring (second generation) with
genotype AA. ...
Get solution
6.2.12
Compute the following probabilities for a selfing plant using Figure
6.2.9. The fraction of third-generation offspring with genotypeAA.
...
Get solution
6.2.13
Compute the following probabilities for a selfing plant using Figure
6.2.9. The fraction of fourth-generation offspring with genotype AA.
...
Get solution
6.2.14
Compute the following probabilities for a selfing plant using Figure
6.2.9. The fraction of offspring with genotype AA in generation t...
Get solution
6.2.15
Suppose
that the fraction of homozygous and heterozygous offspring that survive
self-fertilization by a heterozygote is measured. Find the fraction of
surviving offspring that are heterozygous in the following cases.
All of the homozygous offspring survive, and half of the heterozygous
offspring survive.
Get solution
6.2.16
Suppose
that the fraction of homozygous and heterozygous offspring that survive
self-fertilization by a heterozygote is measured. Find the fraction of
surviving offspring that are heterozygous in the following cases. Half
of the homozygous offspring survive, and one third of the heterozygous
offspring survive.
Get solution
6.2.17 Consider
the following case of blending inheritance. A population of plants
starts out with an equal number of individuals of heights 40 and 60 cm.
The parents mate randomly, and the height of an offspring is exactly
equal to the average height of its parents. What are all the
possible matings? What would be the heights of the offspring? What is
the probability that a 40-cm tall plant mates with a 40-cm tall plant?
Get solution
6.2.18 Consider
the following case of blending inheritance. A population of plants
starts out with an equal number of individuals of heights 40 and 60 cm.
The parents mate randomly, and the height of an offspring is exactly
equal to the average height of its parents. What is the
probability that a 60-cm tall plant mates with a 60-cm tall plant, and
the probability that a 40-cm tall plant mates with a 60-cm tall plant?
Get solution
6.2.19
Consider
the following case of blending inheritance. A population of plants
starts out with an equal number of individuals of heights 40 and 60 cm.
The parents mate randomly, and the height of an offspring is exactly
equal to the average height of its parents. Suppose that these
offspring now mate with each other. Find all possible matings and the
resulting offspring heights.
Get solution
6.2.20 Consider
the following case of blending inheritance. A population of plants
starts out with an equal number of individuals of heights 40 and 60 cm.
The parents mate randomly, and the height of an offspring is exactly
equal to the average height of its parents. Find the probability
of each of the possible matings of the offspring. Out of 100 plants,
about how many would have height 50 cm?
Get solution
6.2.21 Consider
the following case of blending inheritance. A population of plants
starts out with an equal number of individuals of heights 40 and 60 cm.
The parents mate randomly, and the height of an offspring is exactly
equal to the average height of its parents. With blending
inheritance, the height of the offspring is equal to the average height
of the four grandparents. Find the probability that all four
grandparents have height 40 cm and thus the probability that a plant in
the second generation has height 40 cm.
Get solution
6.2.22 Consider
the following case of blending inheritance. A population of plants
starts out with an equal number of individuals of heights 40 and 60 cm.
The parents mate randomly, and the height of an offspring is exactly
equal to the average height of its parents. Find all the ways that the heights of the grandparents average to exactly 50 cm.
Get solution
6.2.23 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. A certain highly mutable gene has a 1.0%
chance of mutating each time a cell divides. Suppose that there are 15
cell divisions between each pair of generations. What is the chance that
the gene mutates in one generation, during the course of those 15 cell
divisions? If there were 100 such genes, about how many would have
mutated in one generation?
Get solution
6.2.24 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. A herd of lemmings is standing at the top
of a clif
f. Each jumps off with probability 0.2 each hour. What is the
probability that a particular lemming remains on top of the cliff after 3
hours? If 5000 lemmings are standing around on top of the cliff, about
how many will remain after 3 hours?
Get solution
6.2.25 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. A molecule has a 5.0% chance of binding to
an enzyme each second and remains permanently attached thereafter. If
the molecule starts out unbound, find the probability that it is bound
after 10 seconds. How long would it take for the molecule to have bound
with probability 0.95?
Get solution
6.2.26 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. In tropical regions, caterpillars suffer
extremely high parasitism, sometimes as high as 15% per day. In other
words, a caterpillar is attacked by a parasitoid with probability 0.15
each day. If a caterpillar takes 25 days to develop, what is the
probability it survives? If a female lays 50 eggs, about how many would
survive? How much lower would the parasitism rate have to be for 2 out
of the 50 caterpillars to survive?
Get solution
6.2.27 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. Suppose that each mutant gene in Exercise
23 has a 1.0% chance of mutating back to the original type each cell
division. Use the Markov chain approach to find the fraction of mutant
genes after 15 cell divisions. How much difference does the correction
mechanism make?
Get solution
6.2.28 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. Suppose that the lemmings in Exercise 24
can sometimes crawl back up the clif
f. In particular, suppose that a
lemming at the bottom of the cliff climbs back up with probability 0.1
each hour. What is the probability that a particular lemming is on top
of the cliff after 3 hours? If 5000 lemmings are standing around on top
of the cliff to begin with, about how many will be there after 3 hours?
How much difference does crawling back up make?
Get solution
6.2.29 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. Suppose that bound molecules in Exercise 25
have a 2.0% chance of unbinding from the enzyme each second. Find the
fraction of molecules that are bound in the long run. What is the
probability that a molecule is bound after 10 seconds?
Get solution
6.2.30 In
each of the following circumstances, find the discretetime dynamical
system describing the probability, find the solution, and use it to
answer the question. Suppose that the caterpillars in Exercise
26 have some chance of eliminating their attacker, thus becoming a
caterpillar again. In particular, suppose that a caterpillar has a 0.03
chance of eliminating a parasitoid each day. Find the probability that a
caterpillar is a caterpillar after 25 days. Is a 3% recovery rate
enough for about 2 out of 50 eggs to end up as caterpillars?
Get solution
6.2.31
In
normal plants, the probability that an offspring of a heterozygous
parent is heterozygous is 0.5. If the survival of heterozygous offspring
differs from that of homozygous offspring, the probability that a
surviving offspring is heterozygous may not be equal to 0.5. For the
following values of the probability, write a discrete-time dynamical
system for the fraction of heterozygous offspring over time, find the
solution, and compute the fraction that will be heterozygous after ten
generations. How does this compare with the fraction for a normal plant?
The probability that an offspring is heterozygous is 0.6.
Get solution
6.2.32
In
normal plants, the probability that an offspring of a heterozygous
parent is heterozygous is 0.5. If the survival of heterozygous offspring
differs from that of homozygous offspring, the probability that a
surviving offspring is heterozygous may not be equal to 0.5. For the
following values of the probability, write a discrete-time dynamical
system for the fraction of heterozygous offspring over time, find the
solution, and compute the fraction that will be heterozygous after ten
generations. How does this compare with the fraction for a normal plant?
The probability that an offspring is heterozygous is 0.4.
Get solution
6.2.33
In
normal plants, the probability that an offspring of a heterozygous
parent is heterozygous is 0.5. If the survival of heterozygous offspring
differs from that of homozygous offspring, the probability that a
surviving offspring is heterozygous may not be equal to 0.5. For the
following values of the probability, write a discrete-time dynamical
system for the fraction of heterozygous offspring over time, find the
solution, and compute the fraction that will be heterozygous after ten
generations. How does this compare with the fraction for a normal plant?
The probability that an offspring is heterozygous is 0.2.
Get solution
6.2.34
In
normal plants, the probability that an offspring of a heterozygous
parent is heterozygous is 0.5. If the survival of heterozygous offspring
differs from that of homozygous offspring, the probability that a
surviving offspring is heterozygous may not be equal to 0.5. For the
following values of the probability, write a discrete-time dynamical
system for the fraction of heterozygous offspring over time, find the
solution, and compute the fraction that will be heterozygous after ten
generations. How does this compare with the fraction for a normal plant?
The probability that an offspring is heterozygous is 0.9.
Get solution
6.2.35
A heterozygous plant with genotype Aa self-pollinates. Find the
probability that an offspring is tall for the following genetic systems.
Only plants that have two A alleles are tall (the allele A is
recessive).
Get solution
6.2.36
A heterozygous plant with genotype Aa self-pollinates. Find the
probability that an offspring is tall for the following genetic systems.
Plants that have either one or two A alleles are tall (the allele A is
dominant).
Get solution
6.2.37
A heterozygous plant with genotype Aa self-pollinates. Find the
probability that an offspring is tall for the following genetic systems.
Heterozygous plants are tall, and homozygous plants are short.
Get solution
6.2.38
A heterozygous plant with genotype Aa self-pollinates. Find the
probability that an offspring is tall for the following genetic systems.
Half of heterozygous plants and one fourth of homozygous plants are
tall (it depends on their position in the greenhouse).
Get solution
6.2.39 A heterozygous plant with genotype Aa self-pollinates,
and then its offspring also self-pollinate. Find the probability that
the offspring of the offspring are tall for the following genetics
systems. Only plants that have two A alleles are tall.
Get solution
6.2.40 A heterozygous plant with genotype Aa self-pollinates,
and then its offspring also self-pollinate. Find the probability that
the offspring of the offspring are tall for the following genetics
systems. Plants that has either one or two A alleles are tall.
Get solution
6.2.41 A heterozygous plant with genotype Aa self-pollinates,
and then its offspring also self-pollinate. Find the probability that
the offspring of the offspring are tall for the following genetics
systems. Heterozygous plants are tall, and homozygous plants are short.
Get solution
6.2.42 A heterozygous plant with genotype Aa self-pollinates,
and then its offspring also self-pollinate. Find the probability that
the offspring of the offspring are tall for the following genetics
systems. Half of heterozygous plants and one fourth of
homozygous plants are tall (it depends on their position in the
greenhouse).A heterozygous
Get solution
6.2.43
Often
geneticists want to change one allele in an outcrossing organism while
keeping the rest of the genome the same. For example, they might wish to
take a specially designed stock of flies and alter the eye color from
red to white. Suppose that the white-eye allele is dominant, meaning
that flies with one or two white-eye alleles will have white eyes. One
procedure used is to take a white-eyed fly and cross it with the
red-eyed stock. The white-eyed offspring are then considered to be the
first generation, and are crossed with the red-eyed stock. Their
white-eyed offspring are considered to be the second generation, and are
again crossed with the red-eyed stock, and so forth. The special
red-eyed stock is homozygous for the desirable allele A at some other
locus, but the white-eyed fly is homozygous for the inferior a allele at
that locus. What is the genotype at the eye color locus in the first
generation?
Get solution
6.2.44
Often
geneticists want to change one allele in an outcrossing organism while
keeping the rest of the genome the same. For example, they might wish to
take a specially designed stock of flies and alter the eye color from
red to white. Suppose that the white-eye allele is dominant, meaning
that flies with one or two white-eye alleles will have white eyes. One
procedure used is to take a white-eyed fly and cross it with the
red-eyed stock. The white-eyed offspring are then considered to be the
first generation, and are crossed with the red-eyed stock. Their
white-eyed offspring are considered to be the second generation, and are
again crossed with the red-eyed stock, and so forth. The special
red-eyed stock is homozygous for the desirable allele A at some other
locus, but the white-eyed fly is homozygous for the inferior a allele at
that locus. What is the genotype at the eye color locus in the second
and subsequent generations?
Get solution
6.2.45
Often
geneticists want to change one allele in an outcrossing organism while
keeping the rest of the genome the same. For example, they might wish to
take a specially designed stock of flies and alter the eye color from
red to white. Suppose that the white-eye allele is dominant, meaning
that flies with one or two white-eye alleles will have white eyes. One
procedure used is to take a white-eyed fly and cross it with the
red-eyed stock. The white-eyed offspring are then considered to be the
first generation, and are crossed with the red-eyed stock. Their
white-eyed offspring are considered to be the second generation, and are
again crossed with the red-eyed stock, and so forth. The special
red-eyed stock is homozygous for the desirable allele A at some other
locus, but the white-eyed fly is homozygous for the inferior a allele at
that locus. What fraction of flies will have the a allele (at the
second locus) after one generation?
Get solution
6.2.46
Often
geneticists want to change one allele in an outcrossing organism while
keeping the rest of the genome the same. For example, they might wish to
take a specially designed stock of flies and alter the eye color from
red to white. Suppose that the white-eye allele is dominant, meaning
that flies with one or two white-eye alleles will have white eyes. One
procedure used is to take a white-eyed fly and cross it with the
red-eyed stock. The white-eyed offspring are then considered to be the
first generation, and are crossed with the red-eyed stock. Their
white-eyed offspring are considered to be the second generation, and are
again crossed with the red-eyed stock, and so forth. The special
red-eyed stock is homozygous for the desirable allele A at some other
locus, but the white-eyed fly is homozygous for the inferior a allele at
that locus. What fraction of flies will have the a allele (at the
second locus) after two generations?
Get solution
6.2.47
Often
geneticists want to change one allele in an outcrossing organism while
keeping the rest of the genome the same. For example, they might wish to
take a specially designed stock of flies and alter the eye color from
red to white. Suppose that the white-eye allele is dominant, meaning
that flies with one or two white-eye alleles will have white eyes. One
procedure used is to take a white-eyed fly and cross it with the
red-eyed stock. The white-eyed offspring are then considered to be the
first generation, and are crossed with the red-eyed stock. Their
white-eyed offspring are considered to be the second generation, and are
again crossed with the red-eyed stock, and so forth. The special
red-eyed stock is homozygous for the desirable allele A at some other
locus, but the white-eyed fly is homozygous for the inferior a allele at
that locus. What fraction of flies will have the a allele (at the
second locus) after t generations?
Get solution
6.2.48
Often
geneticists want to change one allele in an outcrossing organism while
keeping the rest of the genome the same. For example, they might wish to
take a specially designed stock of flies and alter the eye color from
red to white. Suppose that the white-eye allele is dominant, meaning
that flies with one or two white-eye alleles will have white eyes. One
procedure used is to take a white-eyed fly and cross it with the
red-eyed stock. The white-eyed offspring are then considered to be the
first generation, and are crossed with the red-eyed stock. Their
white-eyed offspring are considered to be the second generation, and are
again crossed with the red-eyed stock, and so forth. The special
red-eyed stock is homozygous for the desirable allele A at some other
locus, but the white-eyed fly is homozygous for the inferior a allele at
that locus. How many back-crosses would be necessary to purge 99.9999%
of the inferior genes from the white-eyed fly?
Get solution
6.2.49
One force that can alter the ratio of heterozygotes produced by a
selfing heterozygote is meiotic drive. This means that one allele, say
A, pushes its way into more than half of the gametes (ovules or pollen).
Suppose meiotic drive affects the pollen only and that 80% of the
pollen grains from a heterozygote carry the A allele. Ovules are normal,
and 50% of them carry the A allele. What fraction of offspring from a
selfing heterozygote will be heterozygous?
Get solution
6.2.50
One force that can alter the ratio of heterozygotes produced by a
selfing heterozygote is meiotic drive. This means that one allele, say
A, pushes its way into more than half of the gametes (ovules or pollen).
Suppose meiotic drive affects both pollen and ovules and that 80% of
the pollen grains and ovules from a heterozygote carry the A allele.
What fraction of offspring from a selfing heterozygote will be
heterozygous?
Get solution
6.2.51
One force that can alter the ratio of heterozygotes produced by a
selfing heterozygote is meiotic drive. This means that one allele, say
A, pushes its way into more than half of the gametes (ovules or pollen).
Suppose meiotic drive affects both pollen and ovules but that 80% of
the pollen grains carry the A allele while 80% of ovules carry the a
allele. What fraction of offspring from a selfing heterozygote will be
heterozygous?
Get solution
6.2.52
One force that can alter the ratio of heterozygotes produced by a
selfing heterozygote is meiotic drive. This means that one allele, say
A, pushes its way into more than half of the gametes (ovules or pollen).
How
many generations would it take before the probability of a descendent
of a plant described in Exercise 51 would have less than a 0.01 chance
of being a heterozygote? Compare this with the number of generations
required in the absence of meiotic drive.
Get solution
6.2.53
Heterozygosity
in inbreeding organisms can be restored by mutation. Suppose that
mutations always create brand new alleles. Suppose that each parental
allele has a probability 0.01 of mutating. Suppose first that the
parent has genotype AA. What is the probability that the allele that
came from the pollen is type A? What is the probability that the allele
that came from the ovule is type A?
Get solution
6.2.54
Heterozygosity
in inbreeding organisms can be restored by mutation. Suppose that
mutations always create brand new alleles. Suppose that each parental
allele has a probability 0.01 of mutating. The probability that both
alleles in the offspring are type A is the product of the probability
that the allele from the pollen is A and the probability that the allele
from the ovule is A (we
will derive this in Section 6.5). What is the probability that the
offspring of a homozygous parent is homozygous? What is the probability
that the offspring of a homozygous parent is heterozygous?
Get solution
6.2.55
Heterozygosity
in inbreeding organisms can be restored by mutation. Suppose that
mutations always create brand new alleles. Suppose that each parental
allele has a probability 0.01 of mutating. Suppose a plant is
heterozygous with genotype Aa. What is the probability that the allele
that came from the pollen is type A? What is the probability that the
allele that came from the ovule is type A?
Get solution
6.2.56
Heterozygosity
in inbreeding organisms can be restored by mutation. Suppose that
mutations always create brand new alleles. Suppose that each parental
allele has a probability 0.01 of mutating. Find the probability that
the offspring is AA. Find the probability that the offspring is aa.
What is the probability that the offspring of a heterozygous parent is
homozygous? What is the probability that the offspring of a heterozygous
parent is heterozygous? How does this compare with the result in the
absence of mutation?
Get solution
6.2.57 Consider a molecule that leaves a cell each minute with probability 0.1. If ... =1 when the molecule is inside at time t and ... =0 if the molecule is outside, ... where q0.9 takes on the value 1 with probability 0.9 and 0 with probability 0.1. Define an updating function F and replicate the computer experiment 10 times. Count up how many times the molecule is inside at t =5. The probability mt that the molecule is inside at time t has updating function f ( m )=0.9m. Plot the solution of this deterministic discrete-time dynamical system with ....
Compare the mathematically expected fraction with the fraction you
counted in the stochastic version. What could you do to make the
fraction end up closer to the probability?
Get solution
6.2.58
We can simulate a Markovian molecule that has a 20% chance of jumping
back in with the updating function ... The probability m that the molecule is inside follows the related updating function g ( m )=0.9m + 0.2(1 − m). Plot a solution of the probability equation and a simulation of a single molecule starting from M =m =1
(use enough steps to see what is going on). Solve for the equilibrium
probability. Why doesn’t the simulation seem to approach an equilibrium?
What do the two curves have to do with each other?
Get solution
6.2.59 Plants
with many genes affecting a trait can be simulated by adding up the
effects of each gene. For example, suppose that each of 20 genes adds 1
to the height with probability 0.5 and 0 with probability 0.5. The total
height is the sum of 20 such numbers. Find a way to create 100 such
plants. Which height is most common? What is the largest plant? What is
the smallest plant?
Get solution
