A slowly evolving host moves first in symbiotic interactions
James A. Damore and Jeff Gore
Supplemental Information
Contents
1 Introduction
1
2 Supplemental Methods
2.1 Model details . . . . . . . . . . . . . . .
2.2 Infinite RER model: without migration
2.3 Infinite RER model: with migration . .
2.4 Zero RER limit . . . . . . . . . . . . . .
2.4.1 Horizontal transmission . . . . .
2.4.2 Vertical transmission . . . . . . .
2.4.3 Position based . . . . . . . . . .
2.5 The weak selection limit . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
2
2
2
3
3
3
3
3 Supplemental Figures
3.1 Robustness . . . . . . . . .
3.2 Standard Games . . . . . .
3.3 Iterated Prisoner’s Dilemma
3.4 Host-Symbiont-Phage . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4
4
5
6
7
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction
In this supplemental information, we first give a more detailed description of the model and analytic approximations of the infinite and zero relative evolutionary rate (RER) limit. We then go through some of
the calculations mentioned in the main text. The first supplemental figure tests the robustness of our results to changes in the parameters. Supplementary Figure 2 illustrates the very general transition from the
Nash equilibria to the sequential game equilibrium as RER increases. We also found that our model can be
applied to the popular iterated prisoner’s dilemma game, a common model of mutualisms (Supplementary
Figure 3). Finally, we allow each symbiont to have its own symbionts, like in human-bacteria-phage systems
(Supplementary Figure 4).
2
2.1
Supplemental Methods
Model details
At each timestep, the chance that a host reproduces is:
Probability(Hosts Reproduce) =
1
1 + RER · #Symbionts per Host
(1)
The new symbiont distribution in the host offspring is taken from a binomial distribution with mean
equal to the fraction of symbionts playing A. For the continuous games, Cournot duopoly competition and
iterated prisoner’s dilemma, the new symbionts strategies are taken from a normal distribution with mean
equal to the average strategy of each symbiont and standard deviation of 0.1. When symbionts divide, the
reproducing symbiont is chosen with probability equal to its fitness divided by the total fitness of every
symbiont in every host.
1
2.2
Infinite RER model: without migration
To find the equilibrium distribution of the hosts and symbionts in the infinite RER limit, we first assume
that each symbiont population within a host is in equilibrium with its host’s strategy. This is valid because
the symbionts reproduce infinitely faster than the hosts. For any given host strategy, si is the state where
i symbionts are playing A, where i can range from 0 to the number of symbionts per host. The equilibrium
symbiont distribution is found by creating a transition matrix, P , where the element in the jth row and
ith column is the probability to transition from si to sj . The transition matrix is thus a tridiagonal N xN
matrix, where N is the number of symbionts per host plus one. As an example, Pi+1,i would be:
Pi+1,i = (1 − µ)
N −i
N −i
N −i
r·i
+µ
(N − i) + r · i N
(N − i) + r · i N
(2)
where N is the number of symbionts per host, µ is the mutation rate, and r is the relative fitness of the
symbionts playing A [1, 2]. Because each row adds up to one and every element is non-negative, it can be
shown that the largest eigenvalue, λ0 , is 1 and every other eigenvalue is strictly between -1 and 1. Also, let
v0 be the eigenvector associated with λ0 . Now, if we start in any state x0 :
P x0 = x1 ⇒ QΛQ−1 x0 = x1
(3)
where Q and Λ are matrices containing all of P ’s eigenvectors and eigenvalues, respectively, and x1 is the
state vector for the next time step. Therefore,
P n x0 = xn ⇒ QΛn Q−1 x0 = xn
(4)
As n increases, all of the eigenvalues besides λ0 = 1 disappear and P ∞ becomes a matrix where each column
is v0 . Because the elements in x0 add up to 1, x∞ = v0 is the equilibrium distribution.
Now, given the symbiont distribution for each host strategy, we can find the average payout of each
host strategy and the same analysis as above can be done to find the transition matrix and equilibrium
distribution of the host strategies. Note that because each symbiont population reaches equilibrium before
its host reproduces, the form of symbiont transmission has no effect in the infinite RER limit and is not
modeled here.
2.3
Infinite RER model: with migration
Migration adds a frequency dependence [3]; rather than just being in equilibrium with its host, each symbiont
population also has to be in equilibrium with the whole host population. In fact, if a symbiont immigrates
every time it reproduces, it is equivalent to a well-mixed environment because the equilibrium distribution
of every symbiont population would be the same, irrespective of the host strategy. For intermediate levels
of migration, we assume the distribution of symbionts is the same in every host playing the same strategy.
Therefore, there are now M N xN transition matrices, where M is the number of hosts plus one and N is the
number of symbionts per host plus one squared. Each element of the M transition matrices is the probability
of going from i symbionts playing A in hosts playing A and j symbionts playing A in hosts playing B to
k symbionts playing A in hosts playing A and l symbionts playing A in hosts playing B. The equilibrium
distribution of symbionts can again be found for each of the M transition matrices, which gives the relative
fitness of a host playing A when there are m other hosts also playing A. For reference, the probability of
going from k symbionts playing A in each of i hosts playing A and n symbionts playing A in the H − i hosts
playing B to k + 1 symbionts playing A in the i hosts playing A and n symbionts playing A in the H − i
hosts playing B is:
i−1
H −i
S−k
(1 − γ)((1 − µ)PAA + µPBA ) + γ(
((1 − µ)PAA + µPBA ) +
((1 − µ)PAB + µPBB ))
S · PT otal
H −1
H −1
(5)
where H and S are the number of hosts and symbionts per host respectively, PAB is the payout of symbionts
playing A in hosts playing B, µ is the mutation rate, and γ is the probability that a symbiont will be born
in a different host than its parent.
2.4
Zero RER limit
Unlike the infinite RER limit which is independent of the form of symbiont transmission, the characteristics
of the zero RER limit are dominated by how the symbionts are acquired in each new host.
2
2.4.1
Horizontal transmission
As RER approaches zero, only the hosts reproduce and the distribution of symbionts in each host is just
taken from a binomial distribution with mean equal to the fraction of symbionts playing A when the host
was born. For small host population sizes, where each host is born relatively recently, the probability for the
total number of symbionts playing A to go from i to i + k is:
Pi→i+k =
S
X
P (n)P (n − k) =
n=k
S X
S n
S
p (1 − p)S−n
pn−k (1 − p)S−(n−k)
n
n−k
(6)
n=k
where S is the number of symbionts per host and p is the fraction of total symbionts playing A. This is
simply the probability of a host being born with n symbionts playing A and replacing a host with n − k
symbionts. As RER approaches zero, the payout of the symbionts does not matter and the symbionts spend
as much time playing A as they do playing B and the hosts simply follow the trend and evolve to play the
best strategy for the current symbiont distribution.
2.4.2
Vertical transmission
When the hosts reproduce much faster than the symbionts, a host with a particular symbiont distribution
will fix in the population before that distribution changes. Therefore, all the hosts have the same symbiont
distribution and the probability that the number of symbionts per host playing A increases from i to i + 1 is
simply:
1 − rh−1
(S − i)2
(S − i)rs i
+µ
·
(7)
Pi→i+1 = Pin one host: i→i+1 · Pthat host f ixes = (1 − µ)
S(S − i + rs i)
S(S − i + rs i) 1 − rh−H
where H and S are the number of hosts and number of symbionts per host, and rs and rh are the relative
fitness of the symbionts playing A and the hosts with i + 1 symbionts playing A, respectively. Because of this
selection on multiple levels, the low RER vertical transmission limit can be used to model the prevalence and
virulence of insertion sequences and plasmids in bacteria where virulent, but highly reproductive insertion
sequences are defecting symbionts in a prisoner’s dilemma with the host and functional genes are cooperators.
2.4.3
Position based
When hosts receive their symbionts from the host that they replace, the symbiont distribution is stagnant
and the host population quickly adapts to it. In particular, only the total number of symbionts playing A in
the population is important because the host population responds to the average symbiont subpopulation.
One can then find the average number of hosts playing A for any given total number of symbionts playing
A, which in turn gives the relative fitness of symbionts playing A. Thus, the final transition matrix is N xN ,
where N is the total number of symbionts plus one. Note that this model assumes full migration, which does
not significantly change the results at very low RER. In the interest of space, exact model details are not
given here.
2.5
The weak selection limit
The main text mentions the fitness advantage necessary for significant in-host adaptation of parasites. In
an HIV-positive individual, the effective population size, NE , is approximately 4,000 and the generation
time is approximately 1 day [4, 5]. Given that humans reproduce about once every 20 years, HIV’s relative
reproductive rate is about 73,000. If we want to know the minimum relative fitness advantage necessary for
the population to reach the sequential game equilibrium, we can apply the same analysis as above:
ln N
ln 4, 000
= RER ⇒
= 73, 000 ⇒ r ≥ 1.001
r−1
r−1
(8)
We would therefore predict that for our results to hold, the minimum relative fitness between two
strains of a virus is only 1.001, which equates to the 0.1% growth advantage referenced in the text.
3
3
Supplemental Figures
3.1
Robustness
Equilibrium fraction
of cooperators
a
Equilibrium fraction
of cooperators
d
b
1
0.5
e
1
1
f
Number of Symbionts
Migration
0
0.1
0.25
0.5
0.75
1
0.5
10
Relative evolutionary rate
Number of Hosts
10
20
40
50
100
200
Mutation Rate
0.001
0.005
0.01
0.05
Symbiont Transmission
Biased Horizontal
Position Based
Vertical
Horizontal
0
0
c
Offset
0.05
0.5
1
2
5
100 1
10
Relative evolutionary rate
10
20
40
50
100
200
100 1
10
100
Relative evolutionary rate
Figure 1: Our model is robust to changes in the parameters. Each graph shows the equilibrium fraction of
cooperating hosts (solid line) and symbionts (dashed line) in the snowdrift game under varying conditions.
As RER increases, the population reaches the sequential equilibrium irrespective of the chosen parameters.
(The standard conditions are: horizontal symbiont transmission, zero migration, mutation rate = 0.01, offset
added to each entry of the payout matrix = 1, number of hosts = number of symbionts per host = 40)
4
Standard Games
Battle of the Sexes
Symbiont
Host
A
A
B
B
(2,1) (0,0)
(0,0) (1,2)
Stag Hunt
Symbiont
Host
A
B
B
(2,2) (0,1)
(1,0) (1,1)
Nash ≠ Sequential
Symbiont
Host
A
A
B
B
(2,2) (0,0)
(3,1) (1,3)
Prisoner's Dilemma
Symbiont
Host
A
A
B
B
(3,3) (0,4)
(4,0) (1,1)
1
64%
2%
95%
2%
2%
32%
0%
3%
49%
3%
94%
3%
3%
45%
0%
3%
0.5
>1
0
1
0.5
0
0
1
0%
2%
93%
3%
1%
97%
0%
4%
0%
2%
0%
2%
2%
96%
2%
96%
0.5
log(Frequency relative to average)
A
RER = 100
RER = 1
Fraction of Hosts Playing A Fraction of Hosts Playing A Fraction of Hosts Playing A Fraction of Hosts Playing A
3.2
0
1
<-1
0.5
0
1
0.5
0
Fraction of Symbionts
Playing A
1
0.5
0
Fraction of Symbionts
Playing A
Figure 2: The population goes to the Nash equilibria when RER = 1 and the sequential game equilibrium
when RER = 100. Note that this is the case even when the the sequential equilibrium is not a Nash
equilibrium. Also, the RER has very little to no effect when the simultaneous and sequential equilibria are
the same, as in the prisoner’s dilemma. All graphs used standard conditions of µ = 0.01, offset = 1, Number
of Hosts = Number of Symbionts per Host = 40, horizontal transmission, and no migration.
5
3.3
Iterated Prisoner’s Dilemma
a
b
Defect
4 1
Average value
Cooperate
C D
3 0
1
0.5
c
0
Average payout
Each interaction is an Iterated
Prisoner's Dilemma using:
3
P Host
Q Host
P Symbiont
Q Symbiont
Each player has two variables:
P = Probability to Cooperate
in response to Cooperate
Q = Probability to Cooperate
in response to Defect
(P,
(P,
(P,
(P,
Q)
Q)
Q)
Q)
=
=
=
=
(1, 1) = Always Cooperate
(0, 0) = Always Defect
(1, 0) = Tit-for-Tat
(⅔, 0) = "Miser"
Host
2
Symbiont
1
0
50
Time (in host generations)
100
Figure 3: Mutualisms can also be modeled as an iterated prisoner’s dilemma as in [7, 8]. Both previous
studies found that the slowly evolving host gained a disproportionate amount of the benefits, but neither
connected this idea to other host-symbiont interactions or to sequential games. (a) Our description of the
iterated prisoner’s dilemma and the payout matrix used for the one-off version. (b) The hosts evolve a
“miserly” strategy that keeps the symbionts cooperating but also defects from time to time. (c) The payout
of the host slowly increases to more than can be achieved with Tit-for-Tat. Note that because of mutational
noise, the hosts do not play optimally. (RER = 100; Number of Hosts = Number of Symbionts per Host =
40; µ = 0.01; the standard deviation of each mutant from its parent is 0.05; offset = 1, which was subtracted
from (c); P and Q were initially uniformly distributed between 0 and 1; each interaction was 100 steps of an
iterated prisoner’s dilemma)
6
Host-Symbiont-Phage
3 Layer Model
c
Fraction cooperating
a
d
b
Volunteer's Dilemma
Payout(Cooperators) = 0.5
If the two other players cooperate:
Payout(Defector) = 1
However, if another player defects:
Payout(Defectors) = 0
1
Phage
0.5
Host
Symbiont
0
1
Symbiont
Average payout
3.4
Host
Phage
0.5
0
100
200
300
400
Time (in host generations)
500
Figure 4: The equilibrium remains sequential for games with more than two players. (a) Our multi-layered
model. Each symbiont has its own symbionts (phage) that replicate faster than both the host and symbiont.
(b) Each host-symbiont-phage triplet plays a game called volunteer’s dilemma. All cooperators always get
a payout of 0.5, but a lone defector receives a payout of 1. If, however, two or more players defect, every
defector gets a payout of 0. (c-d) A representative simulation run with phage. Every player starts out
cooperating and the phages are the first to defect. Soon, however, the symbionts begin to defect, forcing the
phages to cooperate. Ultimately, defecting hosts take over the population with the phages and symbionts
cooperating. In this simulation, the population consisted of 20 hosts, each with 20 symbionts that each had
20 phage. The phages reproduced 10 times as often as the symbionts which in turn reproduced 10 times more
often than the hosts. Hosts acquired horizontally transferred symbionts and phages from the environment
and symbionts received horizontally transferred phages from within the same host. More complicated games
could be shown to discern the second and third move, but this game was chosen for its simplicity.
7
References
[1] Moran PAP (1962) The statistical processes of evolutionary theory, (Clarendon Press, Oxford).
[2] Nowak MA (2006) Evolutionary Dynamics: Exploring the Equations of Life, (Harvard University Press,
Cambridge).
[3] Levin BR (1988) Frequency-Dependent Selection in Bacterial Populations. Phil. Trans. R. Soc. Lond. B
319:459–472.
[4] Seo TK, Thorne JL, Hasegawa M, Kishino H (2002) Estimation of effective population size of HIV-1
within a host: a pseudomaximum-likelihood approach. Genetics 160:1283–1293.
[5] Rodrigo AG, et al. (1999) Coalescent estimates of HIV-1 generation time in vivo. Proc. Natl. Acad. Sci.
USA 96:2187–2191.
[6] Herre EA, Knowlton N, Mueller UG, Rehner SA (1999) The evolution of mutualisms: exploring the
paths between conflict and cooperation. Trends Ecol. Evol. 14:49–53.
[7] Frean MR, Abraham ER (2004) Adaptation and enslavement in endosymbiont-host associations. Phys.
Rev. E 69:51913-1–6.
[8] Doebeli M, Knowlton N (1998) The evolution of interspecific mutualisms. Proc. Natl. Acad. Sci. USA
95:8676–8680.
8