Vertical & Oblique Transmission
under Fluctuating Selection
:
Yoav Ram
October 31, 2017
:
Work with Marc Feldman & Uri Liberman
Photo by Kelly Sikkema on Unsplash
Slide 2
Slide 2 text
Who I am
» Yoav Ram (www.yoavram.com)
» Postdoc at Feldman lab, Stanford University
» PhD in mathematical biology at Hadany lab, Tel-
Aviv University
» Evolution of the mutation and recombination rate
» Estimating fitness in microbes
» Also: Python training for engineers & data
scientists (python.yoavram.com)
2/45
Slide 3
Slide 3 text
The Modern Synthesis
Genetic inheritance as the transmission mode of
traits between generations.
> Genetic changes that improve the fitness of
individuals will tend to increase in frequency over
time.
-- Evolution, Bergstrom and Dugatkin 2012, ch. 1.1
Considered to be vertical: parent to offspring
3/45
Slide 4
Slide 4 text
Non-genetic
inheritance
» Cultural evolution:
Imitation, learning...
» Epigenetics
» Microbiome: symbionts,
parasites, pathogens
» Prions: infectious proteins
such as [Het-s],[PSI+],
[URE3], spongiform
Often non-vertical
4/45
Vertical vs. non-
vertical
transmission
Differences in
» persistence
» reversibility
» speed
» timing
» direction
» regulation
Photo by William Bout on Unsplash 6/45
Slide 7
Slide 7 text
Oblique transmission
Offspring inherit traits from
non-parental adults.
*Oblique = Diagonal
Photo by Samuel Zeller on Unsplash 7/45
Slide 8
Slide 8 text
Focus: mixed vertical & oblique transmission
Offspring inherit traits either from parent or from
non-parental adults.
Examples:
» Social learning in humans, birds, dolphins...
» Microbiome
» Genetic inheritance + horizontal gene transfer
8/45
Slide 9
Slide 9 text
Model
Phenotypes
Two phenotypes that affect
fitness:
phenotype A B
freq.
fitness
9/45
Slide 10
Slide 10 text
Model
Transmission
Offspring inherit phenotype from:
» parent with probability
» random non-parental adult with probability
where is the vertical transmission rate.
10/45
Slide 11
Slide 11 text
Model
Recurrence equation
The frequency of phenotype in the next generation:
Mean fitness:
11/45
Slide 12
Slide 12 text
12/45
Slide 13
Slide 13 text
Constant
environment
Result 1. If is favored by
natural selection over and
*, fixation of phenotype
is globally stable.
* With perfect oblique transmission ( ) or neutral
evolution ( ) the recursion is and there is no
change in frequency.
Photo by Amy Humphries on Unsplash 13/45
Slide 14
Slide 14 text
14/45
Slide 15
Slide 15 text
Periodic environment
15/45
Slide 16
Slide 16 text
Periodic environment
Consider environments that favor for
generations and for generations:
-
-
-
» Fitness of the favored phenotype (whatever it is
at a given time) is
» Fitness of the unfavored phenotype is
16/45
Slide 17
Slide 17 text
Periodic environment
Result 2. If then
fixation of either phenotype
is unstable and a protected
polymorphism exists.
Photo by Scott Webb on Unsplash 17/45
Slide 18
Slide 18 text
Periodic environment
Result 3. For general and , a protected
polymorphism exists if
otherwise fixation of one phenotype is stable.
18/45
Slide 19
Slide 19 text
19/45
Slide 20
Slide 20 text
Periodic environment
A1B1
We saw that when there is a protected
polymorphism.
We can find it for .
Result 4. For A1B1 there is a unique stable
polymorphismx
x is the frequency of at the end of even generations
20/45
Slide 21
Slide 21 text
A1B1
If vertical transmission rate
increases
then stable frequency
decreases
21/45
Slide 22
Slide 22 text
A1B1
So, with vertical
transmission, the frequency
of decreases just before
it is favored again.
That is not good.
22/45
Slide 23
Slide 23 text
A1B1
Indeed, the stable mean
fitness decreases with the
vertical transmission rate:
23/45
Slide 24
Slide 24 text
Evolution of the transmission mode
Can the transmission mode itself evolve?
Photo by Matthew Henry on Unsplash 24/45
Slide 25
Slide 25 text
Modifier model
We model competition between
two modifier alleles:
» with vertical
transmission rate ,
» with vertical
transmission rate .
Photo by Cloudvisual on Unsplash 25/45
Slide 26
Slide 26 text
Modifier model
Pheno-
genotype
frequency
fitness
rate
26/45
Slide 27
Slide 27 text
Modifier model
Recurrence equation
27/45
Slide 28
Slide 28 text
Modifier model
Recurrence equation
28/45
Slide 29
Slide 29 text
Stability analysis
Photo by Austin Neill on Unsplash 29/45
Slide 30
Slide 30 text
Modifier model
» Initial population:
» only with modifier allele
» at equilibrium between phenotypes and ( )
» Now, allele is introduced at a low frequency
Can increase in frequency and invade the
population, or is stable to invasion?
30/45
Slide 31
Slide 31 text
Periodic environment: A1B1
Result 7. A modifier allele with vertical
transmission rate is:
» stable to invasion of allele with rate if ,
» unstable if .
The evolutionary stable rate in therefore and
oblique transmission is likely to evolve.
31/45
Slide 32
Slide 32 text
Periodic environment: A1B1
» and switch between
and every generation.
» Initial resident modifier:
.
» Invaders reduce rate by
.
32/45
Slide 33
Slide 33 text
Reduction principle
Photo by Brendan Church on Unsplash 33/45
Slide 34
Slide 34 text
Reduction principle for vertical transmission
In the selection regime, evolution tends to
reduce vertical transmission and increase oblique
transmission.
34/45
Slide 35
Slide 35 text
The plot thickens...
Photo by Cindy Tang on Unsplash 35/45
Slide 36
Slide 36 text
Periodic
environment: AkBk
More generally, there is no
reduction of the vertical
transmission rate.
With environmental periods
the stable vertical
transmission rate is high
(>0.4).
36/45
Slide 37
Slide 37 text
Periodic
environment: AkBk
Moreover, the stable
transmission rate does not
maximize the geometric mean
fitnessg.
g Geometric average of the population mean fitness over
generations.
37/45
Slide 38
Slide 38 text
Periodic
environment: AkBk
Moreover, the stable
transmission rate does not
maximize the geometric mean
fitnessg.
Reminder: these modifiers are
not neutral as they reduce
effect of selection.
g Geometric average of the population mean fitness over
generations.
38/45
Slide 39
Slide 39 text
Polymorphism
Photo by Tatiana Lapina on Unsplash 39/45
Slide 40
Slide 40 text
Polymorphism
» Constant environment:
polymorphism lasts longer
with oblique transmission.
Photo by Tatiana Lapina on Unsplash 39/45
Slide 41
Slide 41 text
Polymorphism
» Constant environment:
polymorphism lasts longer
with oblique transmission.
» Periodic environment:
polymorphism is maintained
in shorter periods with
oblique transmission
Photo by Tatiana Lapina on Unsplash 39/45
Slide 42
Slide 42 text
Evolution of oblique
transmission
» Rapidly changing
environments favor oblique
transmission.
» Slow and constant
environments favor vertical
transmission.
Photo by Ray Hennessy on Unsplash 40/45
Slide 43
Slide 43 text
Phenotype switching
Several studiesξ assumed
- periodically changing environment
- vertical transmission of phenotype
- phenotype switch by transmission fidelity
- fidelity determined by a genetic modifier
ξ Leigh 1970, Ishii et al. 1989, Jablonka 1996, Kussel & Leibler 2005, King & Masel 2007, Liberman et al.
2011
41/45
Slide 44
Slide 44 text
Phenotype switching
» Switching rate evolves
toward where is the
period lengthμ.
» Environmental statistics
encoded by the modifier.
μ Doesn't work if is large or if selection not
symmetric.
42/45
Slide 45
Slide 45 text
Phenotype switching
With oblique transmission:
» Phenotype switch caused by oblique transmission
rather then transmission errors.
» Environmental statistics encoded by the stable
population phenotype distribution if environmental
changes are frequent and selection is weak.
43/45
Slide 46
Slide 46 text
Acknowledgments
Funding: Stanford Center for Evolution and Human
Genomics
44/45