This document discusses the "hitch-hiking effect", where natural selection at one genetic site can impact variation at a nearby neutral site. It begins by defining the question precisely. It then reviews predictions in the absence of selection and introduces models of how selection changes these, including "classic sweeps" of new beneficial mutations and "soft sweeps" of standing variation. It discusses quantifying the process through the structured coalescent and simulating genealogies conditional on selective trajectories. Finally, it introduces the concept of "pseudo-hitchhiking" which ignores fixation time and models hitchhiking as occurring at a rate.

1. The ‘hitch-hiking’
effect
April 4, 2016

2. Motivation
• You want to sequence some individuals and
identify loci “subject to selection”, in some general,
vague sense.
• To make any progress in terms of theory, you first
have to formalize the question.

3. Define the question
• What is the effect of natural selection at “site A” on
the change in allele frequency of “site B”?
A B
r
“Selected
site”
“Neutral
locus”

4. In the absence of selection
E[ x] = 0“HWE”
V [ x] = x(1 x)
2N“More drift
in small
populations”
E[H] = ✓
1+✓ ; ✓ = 4Neµ
“More variability
in large
populations”
(See any introductory evolution/population genetics text)

5. In the absence of selection
E[⇡] = 2 i
n
n i
n 1 = ✓
Expected mean # differences b/w all pairs of
sequences in a sample of size n.
Tajima (1983) Genetics
ˆ✓ = SPn 1
i=1
Watterson (1975)
Theoretical Pop’n
Biology
Expected # of mutations
in sample of size nE[S] = ✓
Pn 1
i=1
1
i
f(i) = ✓
i ; 1  i < n
Expected # of mutation where
derived state occurs i
times in a sample of size n.
Tajima (1983), but see
Hudson (2015) PLoS One for way
easier derivation.
0
2.5
5
7.5
10
1 2 3 4 5
n = 6; ✓ = 10

6. Figure from Hudson, 1990 “Gene genealogies and the coalescent process”

7. How does selection change
these predictions?
• “Classic sweep” - new mutation, beneficial upon
origin. This is 1, 1+sh, 1+2s.
• “Soft sweep” - neutral or deleterious variant,
becomes beneficial later. “Selection on standing
variation.”
• Polygenic trait. This is quadratic selection based
on deviations from an optimum. Will not cover in
detail. I will make qualitative comments.

8. Define the question
• What is the effect of natural selection at “site A” on
the change in allele frequency of “site B”?
A B
r
“Selected
site”
“Neutral
locus”

9. Classic sweeps: intuition
r = 0
r > 0
Heterozygosity ( = diversity)
is reduced at neutral locus due
to “hitch-hiking”. Magnitude of effect
will depend on ’s’ and ‘r’

10. Hitch-hiking effect of a gene 29
-0008 -0006 -0004 -0002 0 0002 0004 0006 0008
Fig. 2. 4Qao(l —Qoo) is
the final amount of heterozygosity at a locus, when initial
frequencies of o, A are 0-5. The graph here, with N = 106
and s = 0-01, is calculated
from (8).
heterozygosity remained, the gene frequencies would return towards their equi-From Maynard-Smith & Haigh (1974)
(Remember this when
reading Kim and Stephan)

11. Quantifying the process
• Need trajectory of beneficial mutation from
frequency 1/(2N) to 1-1/(2N).
• Need a way to simulate a coalescent on top of that.
• This is the “structured coalescent”, introduced by
Dick Hudson and Norm Kaplan
• See recent Perspective by Barton in Genetics:
http://www.genetics.org/content/202/3/865

12. Trajectories
t,-E=X-‘(l -&),
atisfiesthe differential equation
dx(t)-=sx(t)(l -x(t)),dt
x( t,) = E.
on of this differential equation is
x(t) =
&
E+(l--E)e-““-‘“”
(2)
(3a)
nient to introduce a new variable r = t - t,. The time it takes for
essto go from Eto 1-E is
f = -2 ln(s)/s. (3b)
r to describe the effect of the selected mutation on a linked
cus, Ohta and Kimura (1975) divide the population into two
part consists of chromosomes carrying the advantageous muta-
other one the disadvantageous allele b. Let pi be the frequency of
mong chromosomescarrying the favorable mutation B, and pZthe
of allele A among b-chromosomes.Note that these variables are
rom the usual state space variables of two-locus, two-allele
urthermore, let ~$(pi, p2, T) be the joint probability density func-
and p2 at time t > 0. Our goal is to compute the expectations of
nciesp, and p2 and their second-order momentsp:, p1 pz, and pi
s differential equation is
x(t) =
&
E+(l--E)e-““-‘“” (3a)
introduce a new variable r = t - t,. The time it takes for
o from Eto 1-E is
f = -2 ln(s)/s. (3b)
scribe the effect of the selected mutation on a linked
a and Kimura (1975) divide the population into two
nsists of chromosomes carrying the advantageous muta-
e the disadvantageous allele b. Let pi be the frequency of
omosomescarrying the favorable mutation B, and pZthe
A among b-chromosomes.Note that these variables are
usual state space variables of two-locus, two-allele
e, let ~$(pi, p2, T) be the joint probability density func-
Deterministic.
From Stephan et al. (1992, TPB)
itesimal mean includes an additional term, which effec-
y gives the appropriate push toward the boundary on
ch we have conditioned.
ur approach also relies on the reversibility of the diffu-
process (cf. Griffiths 2003). Specifically, we use the fact
the diffusion process looking backward in time from the
ent (i.e., toward the introduction of the allele) has the
e distribution as a process forward in time conditional
bsorption at zero. This conditional process (t) is theX*N
e as XN(t) but with ␮N(x) replaced by (x) ϭ Ϫx (Ewens␮*N
4). Likewise, because we are only interested in beneficial
es that eventually reach fixation, we consider the dif-
on process conditional on the selected allele reaching a
uency of one. This conditional process (t) has an in-ϩ
XS
esimal mean (x) ϭ 2Nsx(1 Ϫ x)/tanh(2Nsx) (Ewensϩ
␮S
4).
o generate a trajectory for allele A, we use a variable-
d jump random walk to approximate to the diffusion pro-
. Given a current frequency x, at time intervals ⌬t, the
uency x jumps to either:
x → x ϩ ␮(x)⌬t Ϫ ͙x(1 Ϫ x)⌬t or (2a)
x → x ϩ ␮(x)⌬t ϩ ͙x(1 Ϫ x)⌬t (2b)
equal probability. The term ␮(x) is replaced by the con-
onal infinitesimal mean of the phase in question (i.e.,
ral or selective). This process has the correct diffusion
t, that is, the correct infinitesimal mean and variance are
ined and all higher moments are zero, as the time interval
→ 0 (Karlin and Taylor 1981). Hence, for small ⌬t, it
ides a good approximation to the diffusion process. We
fied this for our choice of ⌬t ϭ 1/(4N) by comparison to
ytical expectations and to alternative methods of simu-
variation, we simulate samples from a linked, neutrally evo
ing region using a structured coalescent approach. Spec
cally, we generate a trajectory of allele A from introduct
to fixation, then condition on this particular realization of
genealogical process to generate an ancestral recombinat
graph for our sample (Fig. 1). The trajectory of allele A
modeled stochastically, using a new approach (see Method
Under the standard sweep model, f ϭ 1/(2N), while un
the model of directional selection on standing variation, f
1/(2N).
Effect of f on Diversity Levels
Irrespective of the value of f, mean diversity levels
most distorted near the selected site and tend toward th
neutral expectation with increasing genetic distance. Thi
illustrated in Figure 2A, using parameters that may be
plicable to humans (e.g., Frisse et al. 2001). We present th
summaries of diversity: ␪W (Watterson 1975), ␪H (Fay a
Wu 2000), and ␲ (Tajima 1989). Under a neutral equilibri
model, these statistics provide an unbiased estimate of ␪,
population mutation rate (␪ ϭ 4N␮, where ␮ is the mutat
rate per generation per base pair). For these parameters
standard sweep leads to a reduction in the mean levels
variation throughout the 100-kb region (relative to the neu
expectation of ␪ ϭ 0.001 per base pair).
A very similar picture is expected so long as f Ͻ 1/(2
and selection is strong (Stephan et al. 1992). As an examp
in Figure 2A the expected levels of variation are indis
guishable for f ϭ 1/(2N) ϭ 5 ϫ 10Ϫ5 and f ϭ 1/(2Ns) ϭ 10
As f increases, the substitution of a favored allele has a we
er effect on diversity at linked neutral sites (Innan and K
2004); for f ϭ 0.20, the effect is hardly detectable. If
Stochastic. From Przeworski et al.
(2005), orig. Coop & Griffiths (2004) TPB
but kinda hard to dig out.
TL;DR - the latter is preferred. The former over-estimates
time to fixation by approximately two-fold.

13. Kaplan et al.Hudson and C. H. Langley
ly
he
ly
dy
bi-
he
1-E
X
E
past present
Pr(escape) ⇡ r/s

14. Kaplan et al.896 N. L. Kaplan,R. R. Hudsonand C. H. Langley
10'
10'
4 3 2 1
10 10
1
10
a
10 1010
I
c
I
I
insensitive to 2N so long as a
not shown).
The major goal of this pa
consequence of hitchhiking r
selected substitutions on stan
variation at the DNA level. I
tion la), is plotted as afunc
values of a with 2N = 10'. S
to 2N (for fixed a),the same
smallvaluesof A, the hitchh
E ( T ) substantially from 2 (it
lated, selectively neutral locu
e.g., if a 3 lo5, and 0.0002 <
G 0.7. Since theexpectednum
sites, E(S), is proportional to
hitchhiking effect associated w
selected mutants (or very rare
reduce the expected number
E[T] = E[total time on tree]
Var.reduced
Recent sweep Old sweep
Var.notreduced
Strong
selection!!!!
↵ = 2Ns
R = 2Nr
⌧ = Generations since fixation
2N

15. Kaplan et al.
10'
10'
10'
10'
10'
10'
4 3 2 1
10 10
1
10
a
10 1010
I
c
I
I
1
2
J
4
l-
l o 4
3
a=10
10-2 10 -l loo 10'
FIGURE3.--E(T), theexpected size (measured in 2N genera-
tions) of the ancestral tree of a sample of two genes at a selectively
to 2N (fo
smallvalu
E ( T ) subs
lated, sele
e.g., if a 3
G 0.7. Si
sites, E(S
hitchhikin
selected m
reduce th
a sample
tion.
The ex
region ofs
ancestor o
Equation
A M A X ran
lo6). It is
near them
about 5 a

16. s/r
• Routinely mis-quoted as “distance at which a
sweep will affect variation”.
• Wrong! It is distance at which site has Pr(escape)
close to 1.

17. an
ed
of
=
ple
ed.
the population (or when the rareallele becomes selec-
tively favored). In Figure 3a, a = lo4 and in Figure
3b, R = 10. It is not difficult to show from Figure 2
and Equation (12) that E(T)is an increasing function
of r and R and decreasingfunction of a.In particular,
as is seen in Figure 3, a and b, E(T) will differ
plotted against T, the ancestral time of fixation of theselected
substitution, a. For different values of R, the expected number of
crossovers between the neutral region and theselected locus per
genome per 2N generations (a= lo4),and b. For different values
of selection, a (R = 10);(see text for explanation).
significantly from 2, its neutral value, if 7 < 0.1 and
R / a < 0.01. This means thattheexpected level of
variation will be substantially reduced for all the sites
within a physical distance of (O.Ol)a/C base pairs of a
locus at which a selected substitution has recently
occurred.Forexample, if 2N = lo8,s = and c =
thenthe width of the affectedregion is only
about 200 bp. But if s = and c = 10-', thenthe
expected variation is reduced in a region about 2000
bp wide.
In Table 2 the values of M (Equation 20), A M A X
(Equation 22)and Z22(M) are given for differentvalues
of a (2N = lo8and 6 = 0.01). The value of Mf in (20)
m
th
M
po
th
w
pr
an
co
cr
se
m
th
dy
th
ef
ar
(Remember this when reading Kim and Stephan)

18. Which SFS?? (Blue = no selection, for
reference)
0
5
10
15
20
1 2 3 4 5 6 7 8 9
Single recent,
strong sweep.
Fay and Wu.
Sweeps occurring at some rate.
Braverman et al., Przeworski.

19. Hitchhiking Effect 789
0.0
-0.5
Q
- -1.0
v)
.-E
p
a
0)
9
a
2 -1.5
-2.0
-2.5
I -
-a= a = 10 lo”
a = lo3
0.0000 0.00050 0.0010
4
over by these typical 6s gives &. Table 2 lists & for
FIGURE4.-Theaveragevalue
of Tajima’s D as a function of A,.
Theparametersare n = 50, S =
17, a = lo’, lo4, lo5, lo6,or lo7,
and A,. ranges from zero to AMx,
which varies depending on a.
0.0015
2). If one takes this as an estimate of &.), then ac-Fig. from Braverman et al. (1995) Genetics

20. “Pseudo-hitchhiking”
• Ignore the trajectory—assume fixation time close to
zero. This means assuming very strong selection.
• Hitch-hiking events occur at rate rho, at which time
all lineages coalesce (in absence of
recombination).

21. “Pseudo-hitchhiking”
E[ x] = 0
V [ x] = ⇢x(1 x)
⇢ = Rate of hitch hiking events
Ne = N
1+2N⇢y2 ! 1
⇢y2 as N ! 1
The extent to which “rho” (and y…) depend on N determines
the extent to which variation is constant across species.

22. J. H. Gillespie
all be lowered
g the various
ur during the
may even be
rs to be quite
randomness,
y.
eudohitchhik-
ainder of this
of the model
y of n alleles
Figur e 7.—The average values of Tajima’s D for different
ation with de- sample sizes. Those E{D(n)}curves come from samples drawn
only way that from a direct simulation of the pseudohitchhiking model for
hhiking event. a sample of size n. The D(n) curves come from a direct simula-
tion of the coalescent using Equation 18. In both cases, r 5icular genera-
0.138, y 5 0.3, and u 5 5 3 1024
.le copyof one
s frequencyto
ed alleles areFigure from Gillespie (2000)

23. Soft sweeps: intuition
Beneficial mutation initially on > 1 genetic background,
leading to prediction that reduction in diversity at linked,
neutral sites will not be as extreme as for a “classic” sweep
(now often called a “hard” sweep).

24. Soft sweep simulation
• Stochastic trajectories
• Neutral trajectory + selected
“stitched” together.
• Vary “ts”, f, etc.
2314 MOLLY PRZEWORSKI ET AL.
FIG. 1. A possible genealogy for six chromosomes at a neutral locus linked to a site where a beneficial alle
In this example, A has just fixed in the population (at time T ϭ 0), so all lineages carry the favored allele. G
is favored from T to ts then neutrally evolving from ts (when it is at frequency f) to tm. The trajectories fo
phases are shown in black and gray, respectively. The coalescent genealogy for the six chromosomes is depict
recombination events between allelic classes are indicated with slanted arrows. Most coalescent events occ
frequency. Because A is neutrally evolving from ts to tm, its sojourn time is longer than it would be under a stan
more opportunity for recombination. Note that, in this example, the most recent common ancestor has not been re
Figure from Przeworski et al. (2005) Evolution

25. FIG. 2. Mean diversity levels as a function of distance from the selected site for different values of f, the fre
is first favored. Diversity levels are summarized by the mean ␲ (dashed), ␪W (gray) and ␪H (black). Under the n
all three statistics are unbiased estimators of ␪, the population mutation rate. (A) Plausible parameters fo
simulations were run for 100 chromosomes, with N ϭ 104, s ϭ 0.05, and ␪ ϭ ␳ ϭ 10Ϫ3 per base pair (␳ ϭ 4
parameter definitions). The time since the fixation of the beneficial allele is zero. Under the neutral equilibr
ϭ E(␪H) ϭ 1 per kilobase. (B) Plausible parameters for Drosophila melanogaster. A total of 104 simulations were
with N ϭ 106, s ϭ 0.01, ␪ ϭ 0.01 per base pair, and ␳ ϭ 0.1 per base pair. The time since the fixation of th
Under the neutral equilibrium model, E(␲) ϭ E(␪W) ϭ E(␪H) ϭ 1 per 100 bp.
has more effect on ␲ than ␪W in all four examples, this is To quantify this observation, we es
f=0.05: reduction in
variation not
nearly as pronounced
f=0.2: effect would be
very hard to detect
“Hard” sweep: variation strongly
reduced near selected site
⇡ = dashed
✓W = grey
✓H = black

26. Take-home
• To detect sweeps, they should be:
• strong (relative to r)
• recent
• in regions of low recombination (relative to s)
• been selected on when rare

27. Quantitative traits
• Any one beneficial mutation not guaranteed to fix.
• I.e., sweeps can “stall out”—see Chevin and
Hopital (2008) Genetics
• B/c genetic background may move mean trait
value to optimum before fixation occurs.
• But, patterns of hitch-hiking should depend mostly
on whether or not mutation was rare or not at onset
of selection.

28. Considerations
• Demographic null model matters. Methods we
read will have to “account for demography”
• How much HH does there need to be before this is
impossible? (This is an open question.)