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WEEK #20: Probabilistic Models, Diffusion and Genetics
Goals:
Textbook reading for Week #20: Study Adler Section 6.1, 6.2.
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Course Information
For the rest of the course, we will be working with fundamental ideas in probability,
and how they relate to biological models, as well as to our earlier work with calculus.
The textbook references for the remaining material comes from the original course
text, “Modeling the Dynamics of Life” by Adler, Chapters 6, 7, and 8.
Week 20 – Probabilistic Models, Diffusion and Genetics
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Introduction To Probabilistic Models
From Section 6.1
Contrast the underlying assumptions of deterministic vs stochastic models.
Give two examples of deterministic population models, one with discrete
time, and one with continuous time.
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Example: A discrete-time stochastic population model could be defined
by the following statements:
• 60% chance a population will have a per capita production of 1.2 in a given
year,
• 20% chance that the production rate will be 1.0 per capita (population
remains the same), and
• 20% chance that the production rate will be 1.4 per capita.
What do you think the average per capita production rate will be under this
model?
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Simulation
For the purposes of comparison, let us define two models, one deterministic and
one stochastic.


1.2Pt 60% chance
Pt+1 = 1.2Pt
Pt+1 = 1.0Pt 20% chance


1.4Pt 20% chance
If we start both models at an initial population of P0 = 1000, can we predict
exactly when each model will reach P = 150, 000?
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For the stochastic or probabilistic model, there is an element of chance to the
future predictions about population. In an era of inexpensive computing, an easy
way to explore the predictions of a model are to simulate them. We program a
computer to simulate a random number being drawn, choose a population rate as a
result of the number, and then update the population according to that randomlyselected rate. Repeating this work over and over, we can predict the population
curve several times to understand the possible range of behaviour in the model.
Week 20 – Probabilistic Models, Diffusion and Genetics
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0
5
10
15
Years
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25
0
100000 200000
+
+
+
+
+
+
+
++
+
+++
+
+
+
++++++++++++++
30
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+
Pop.
100000 200000
0
Pop.
+
+
+
+
+ ++
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+
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+ +
++
+
+
+ ++++
+
+
+ +
++++
+ ++
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++
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+ +
++++
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+++
++++++
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+++
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++++
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++
0
5
10
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Years
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30
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What would be some stochastic alternatives to the deterministic question “At
what time does the population reach 150,000”?
What other questions might be relevant, given the stochastic population model?
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One way to help answer some of the questions about a model’s predictions is
to use the simulation approach to generate a large number of predicted
populations, and then display the range and popularity of various outcomes
using a histogram.
0
5000
15000
First Year The Population Exceeded 150,000
Freq (out of 100,000)
30000
10000
0
Freq (out of 100,000)
Population at Year 30
0
500000
1500000
Population
20
25
30
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Cross Year
40
45
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Which of our earlier questions can be answered based on these histograms?
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As we will see several times in our study of probability, the logarithm transform
is an important tool. Consider the range of year 30 populations we simulated
earlier. The histogram had an odd, asymmetric shape to it. Compare that to the
histogram of the log transformed populations (log base 10):
0
5000 10000
Population at Year 30
Freq (out of 100,000)
30000
10000
0
Freq (out of 100,000)
Population at Year 30
0
500000
1500000
Population
4.5
5.0
5.5
6.0
Log10(Population)
6.5
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On the graph, indicate the original units of the logged populations shown.
Use the histogram to determine (roughly) the average or expected population at year 30.
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Markov Chains
An exciting development in modeling tools for biologists and geneticists in the
last 20 years are Markov Chains or Markov Models. They were studied
heavily in the 1960s, and starting in 1989 they were used to identify different
region types in DNA, RNA, and protein sequences; today they are used to mine
the ever-growing genetic databases for alignments of DNA and RNA sequences
across different organisms for evolutionary comparison, and to identify common
protein sequence and structure motifs for drug development. Markov chains have
also been used in fields of synthetic music generation and authorship identification,
speech recognition and generation, and other sequence classification tasks.1
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Typically, these applications are actually using Hidden Markov Models, or HMMs. HMMs models are a relatively simple extension of the Markov chains we will
study in this course.
http://www.comp.leeds.ac.uk/roger/HiddenMarkovModels/html dev/main.html
http://www.research.ibm.com/journal/rd/453/birney.html
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In eukaryotes, we have long regions of so-called “junk”, or non-coding, DNA. These
sequences are not entirely random, however: certain bases are more likely than
other, and certain short subsequences are more likely than others. If we consider a
simple two-base DNA universe to start with, one way to model these sequences is
with a state diagram like the one shown below:
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Here are some sample sequences from that model
A A A A A A G A G A A A G A G A A G A A A A A A A A G A A A ...
G G A A A G A G A A G A G A G A G A A A A A A G A G A A G A ...
G A A A G A A A G A A A A G A A A A G A A A A A G A A G G A ...
A A G A A A G A G G G G A G G G A A A A A A G G A A G A G G ...
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Question: Which of the following sequences is not likely to have come from this
model?
A) G A G A A G G A G G G G A G G G A G G A A G G G G G G A A G ...
B) A A G G G G G G G A A G G A A G A G G G G G A A A A A G G G ...
C) G A A A A G G G A G A G A G G G A G A G G A A G A A G A A A ...
D) A G A G A G G A G G A A A A A G A G A G A A A A A G A G G A ...
E) A A G A A A G A G A G G A G G A G G G A G A A G A A A A A G ...
Week 20 – Probabilistic Models, Diffusion and Genetics
How sure can you be of your answer?
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Stochastic Models of Diffusion
From Section 6.2
Diffusion is something we have studied in both discrete and continuous time, but in
a deterministic way. In fact, chemists and physicists would argue that diffusion is
more accurately a random or stochastic process, because individual molecules
pass through the membrane based on collision angles, energy levels, steady and
transient polar charge forces, etc.
Sketch:
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However, we can still keep an eye on the process from a macroscopic level, by
measuring concentrations on both sides of the membrane.
Example: By repeated experiments we find that, on average, the concentration of a toxin inside a membrane drops by 20% per hour. Draw a digram for
a stochastic model for an individual toxin molecule over time.
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We now define pt = probability that the molecule is inside the membrane
at time t (hours). If we assume that the molecule is inside the membrane
initially, write the next few values for pt, and then a general formula.
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What kind of equation is this? How does it differ from similar models we have
seen before?
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The discrete time models we have seen before are related to these stochastic ones.
In this case, they are associated by expectation.
Question: After 2 hours, we have p2 = 0.64. This means that at t = 2,
A) there is a 64% chance of a particular molecule still being inside the membrane.
B) exactly 64% of all the molecules we started with will still be inside the membrane
C) approximately 64% of all the molecules we started with will still be inside the
membrane
D) There is a 64% chance that all of the molecules will still be inside the membrane
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Effects of Sample Size
Consider the following simulation graphs.
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5
10
t
15
20
4
3
2
0
1
# Molec. Inside
3
2
0
1
# Molec. Inside
3
2
1
0
# Molec. Inside
Simulation of 4 molecules
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Simulation of 4 molecules
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Simulation of 4 molecules
0
5
10
t
15
20
0
5
10
t
15
20
24
0
5
10
t
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20
40
30
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0
10
# Molec. Inside
30
20
0
10
# Molec. Inside
30
20
10
0
# Molec. Inside
Simulation of 40 molecules
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Simulation of 40 molecules
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Simulation of 40 molecules
0
5
10
t
15
20
0
5
10
t
15
20
Week 20 – Probabilistic Models, Diffusion and Genetics
Simulation of 400 molecules
Simulation of 400 molecules
0
5
10
t
15
20
200
0
100
# Molec. Inside
300
400
400
300
200
0
100
# Molec. Inside
300
200
100
0
# Molec. Inside
400
Simulation of 400 molecules
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0
5
10
t
15
20
0
5
10
15
20
t
What graph you think we’re approaching as we increase the number of molecules
in our simulation?
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Here are similar simulations, but with the function p(t) = p0(0.8)t overlayed for
comparison.
5
10
t
15
20
600
0
0
0
200
# Molec. Inside
80
60
40
# Molec. Inside
20
10
8
6
4
2
0
# Molec. Inside
Simulation of 1000 molecules
1000
Simulation of 100 molecules
100
Simulation of 10 molecules
0
5
10
t
15
20
0
5
10
t
15
20
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We can also view these simulations in tabular form. The values shown here are the
same as those used to generate the graphs above.
t
0
1
2
3
4
5
6
7
8
9
10
pt = 0.8t
1.000 0.800 0.640 0.512 0.409 0.327 0.262 0.209 0.167 0.134 0.107
Sim N0 = 10
10
9
8
7
7
6
4
3
2
1
0
Sim N0 = 100 100
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69
52
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Sim N0 = 1000 1000 805 641 522 401 310 238 191 151 120
92
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You should note that the values of the simulations do not get closer to the
expected values as we use more molecules. e.g. at t = 6, the individual
simulations were off by 1, 3, and 18 molecules respectively for N0 = 10, 100,
and 1000. Why then does the graph for N0 = 1000 look so much closer to the
ideal curve than the smaller sample simulations?
Week 20 – Probabilistic Models, Diffusion and Genetics
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If there are billions of molecules in a sample, what are the trade-offs of using
a deterministic model instead of a stochastic model for diffusion?
Give an example of a scenario or experiment where a deterministic model for
diffusion would not be appropriate.
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Markov Chain Model for Diffusion
So far we have only modeled one-way flow through the membrane. However, our
earlier studies covered the more realistic case of two-way flow through the membrane. We can construct a stochastic model for this using a Markov chain.
Describe the assumptions in this model in words.
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The original scenario, where the molecules could only leave the membrane,
can also be written as a Markov chain. Sketch the transition diagram for our
original model.
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Using the leave-and-enter model, assume a single molecule starts inside the
membrane. Compute the probability that it is in the membrane for the next
few time points.
Week 20 – Probabilistic Models, Diffusion and Genetics
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Find the general formula for pt+1, the probability that an individual molecule
with be inside the membrane at time t + 1. It will depend on pt.
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Find the long-term probability that an individual molecule will be inside the
cell.
Week 20 – Probabilistic Models, Diffusion and Genetics
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Here are some simulations, showing both the ideal number of molecules and simulated number, for different numbers of initial molecules. Note the long-term stable
level indicated in the graphs.
10
20
30
t
40
50
600
0
0
0
200
# Molec. Inside
80
60
40
# Molec. Inside
20
10
8
6
4
2
0
# Molec. Inside
Simulation of 1000 molecules
1000
Simulation of 100 molecules
100
Simulation of 10 molecules
0
10
20
30
t
40
50
0
10
20
30
40
50
t
Comment on the agreement between the ideal probabilities and the simulation
results.
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Genetics
You should read the presentation of selfing and inbreeding effects, pages 498500, on your own. The case of self-reproduction or inbreeding is an interesting
one, especially on its effects on genetic diversity, but we do not have time to
cover it in class.
Genetics are a rich source of probabilistic models. To start with, we will explore
some simple models applied to plant breeding.
Nomenclature
• an allele is a variant of a gene.
• an organism is diploid if it has two separate copies of a gene
• a diploid organism is homozygous if both alleles for a gene are the same,
and heterozygous if the alleles are different.
• The set of alleles of an organism is its genotype. This is usually not directly measured, but inferred from the visible or macroscopic properties of the
organism, its phenotype.
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The presence of different alleles for a set of genes can affect the phenotype of an
organism in a variety of ways.
Give some examples of genotype/phenotype relationships.
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Probability Theory
From Section 6.3
It will be helpful, before we go too much further, to define some nomenclature and
properties of probabilities and experiments.
Sample space for an idealized experiment:
Experiment: Measuring the height of a random student in the class.
Experiment: Throwing a dart at a wall with a dart board on it.
Experiment: Measuring the per-capita reproductive rate for a population over one
year.
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Event in an experiment:
Experiment: Measuring the height of a random student in the class.
Experiment: Throwing a dart at a wall with a dart board on it.
Experiment: Measuring the per-capita reproductive rate for a population over one
year.
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Representing Sample Space and Events
Venn diagrams and line graphs are often used to represent the sample space, events,
and their interrelationships.
Draw a Venn Diagram and line diagram for the annual per-capita reproductive
rate experiment, showing the event “population grew”.
Add to the diagrams the second event “population grew at a rate r = 0.5 or
higher”.
Add to the diagrams the third event “population was constant or shrunk.”
Week 20 – Probabilistic Models, Diffusion and Genetics
Nomenclature
Define the following terms, then give some examples of each.
• Mutually exclusive events
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• Union of two events
Week 20 – Probabilistic Models, Diffusion and Genetics
• Intersection of two events
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• Complement of an event
• Collectively Exhaustive Set of events
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Formal Probability Models
A probability model assigns probabilities to all events. We will write Pr(A)
for the probability of an event A.
To be self-consistent, the assigned probabilities must satisfy the following 4 requirements (or “axioms”).
For each rule, draw a Venn diagram representing it.
1. If S is the the sample space, Pr(S) = 1.
2. 0 ≤ Pr(A) ≤ 1 for any event A.
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3. If A and B are mutually exclusive events, Pr(A ∪ B) = Pr(A) + Pr(B).
4. If Ac is the complement of A, then Pr(Ac) = 1 − Pr(A).
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We see now why our earlier models had certain properties.
Example: For the diffusion model shown below, describe the sub-experiment
that occurs at each step, the sample space for each step, and what constraints
there are on the transition probabilities as a result.
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If we were to run the diffusion model for 10 time steps, how would we define
the sample space?
Give examples of events we could describe based on the sample space for the
10-step diffusion experiment.