Supplementary Material
S1. Use of the Animal Model to estimate genetic variance components in a Daphnia
magna population (Based on Lynch and Walsh, 1998; and Kruuk, 2004)
Description of model and its use
The Animal Model is a form of mixed linear regression model in which the phenotypic values of a
given trait of all observed individuals (the ‘animals’) and the knowledge about the family pedigree
(i.e. the family relationships among all these individuals within the population) can be used to
estimate, for instance, additive and dominance genetic values of each individual for that trait. The
additive genetic value is also known as ‘the breeding value’, referring to its usefulness in for instance
cattle breeding programs, which was historically the first application of the Animal Model. In
addition, this model can also be used to produce estimates of the variance components contributing to
the total phenotypic variance, i.e. the additive genetic variance, the dominance genetic variance and
residual variance. Because the analysis of the Animal Model always begins with the estimation of the
variance components and because in the current paper we are only interested in the estimates of the
variance components of the population (and not in the separate genetic values of each individual), we
limit ourselves to providing a summary of the mathematical details on how the Animal Model is used
to define and estimate the different variance components of interest from the observed data.
According to the Animal Model, observation yi (i.e., in the present paper the phenotype of net
reproductive rate, R0) for each individual i out of a total of k observed individuals is expressed as:
yi    ai  di  ei
(Eq. S1)
where yi is the observed phenotype for individual i (note that each single ‘replicate’ observation of
each clonal lineage in Tables S2 to S25 represents the observation on a different individual); µ is the
population mean (fixed effect), the ai is the additive genetic value of individual i (random effect), di is
the dominance genetic value of individual i (random effect), ei is a residual error (random effect). For
the total set of k observations for k individuals the Animal Model can be written in matrix form:
y = µ + u1 + u2 + e
(Eq. S2)
where y is the (column) vector of observations, u1 and u2 are vectors of random additive and random
dominance genetic effects, respectively, and e is a vector of residual errors; i.e.
𝑦1
𝑎1
𝑒1
𝑑1
𝑦2
𝑎2
𝑒
𝑑
2
𝑦 = ( … ) , 𝑢1 = ( … ) , 𝑢1 = ( 2 ), 𝑒 = ( … )
…
𝑦𝑘
𝑎𝑘
𝑒𝑘
𝑑𝑘
(Eq. S3)
Now, in order to analyze this mixed linear model, and to be able estimate the three variance
components mentioned above, one needs to define the k x k variance-covariance matrices for the
vectors of random effects, i.e. G1, G2, and R, for u1, u2 and e, respectively. G1 and G2 are derived from
expectations of covariance of the additive and dominance genetic effects among relatives,
respectively. These expectations are based on the known family relationship between each pair i, j of
individuals:
G1   a2 A
(Eq. S4)
G2   d2 D
(Eq. S5)
where 𝜎𝑎2 and 𝜎𝑑2 are the additive and dominance genetic variance of R0 in the population,
respectively. The additive genetic relationship matrix A has elements
Aij  2ij
(Eq. S6)
where ij is the coefficient of coancestry for the pair i, j of individuals, which is based on the known
family relationship for each pair i, j of individuals (i.e., 0.5 for clone-mates, 0.25 for parent-offspring,
0.125 for half-sibs, and 0 for unrelated individuals; see further, see also Table 7.1 in Lynch and Walsh,
1998). The dominance genetic relationship matrix D in Eq. S5 has elements
Dij   ij
(Eq. S7)
where ij is the coefficient of fraternity for the pair i, j of individuals, which is also based on the
known family relationship for each pair i, j of individuals; (i.e., 1 for clone-mates, 0 for parentoffspring, 0 for half-sibs, and 0 for unrelated individuals).
For the design of our study the four above-mentioned types of relatedness can be understood as
follows (note that the relationship between any pair of observed individuals is one – and only one - of
these four possibilities):
-
Clone-mate relation: each pair of individuals
within the same clonal lineage (denoted
‘replicates’ in Tables S2 to S25) have a clone-mate relationship to one another;
-
Parent-offspring relation: each individual from a given offspring clonal lineage has a parentoffspring relationship with each individual from either one of the two parental clonal lineages
that was used to produce that offspring lineage by sexual crossing (in terms of Table 1 of the
main text, this type of relation applies to the relation between an individual of an offspring
clone with an individual from its corresponding paternal clone in the most left column or from
its corresponding maternal clone in the upper row; examples of this is marked in Table 1 with
* and **).
-
Half-sib relation: individuals from two different offspring clonal lineages are considered halfsibs if they share only one (not two) of the parental lineages that were used to produce the
offspring clonal lineage (in terms of the compound clone names in Table 1 of the main text,
this type of relation applies to any pair of individuals from two different offspring clonal
lineages which share one of the two letters of their compound clone name; examples of these
are marked in Table 1 with § and §§)
-
Unrelated individuals: individuals which have none of the above three types of relationships
(this applies to pairs of individuals from a different parental clonal lineage and, in terms of
Table 1, to pairs of individuals from offspring clonal lineages that do not share any of the two
letters of their compound name; examples of these are marked in Table 1 with & and &&).
Finally, assuming uncorrelated residual errors among individuals, the variance-covariance matrix for
the residual errors is just:
R   e2 I
(Eq. S8)
where 𝜎𝑒2 is the residual variance of R0 in the population and I is the k × k identity matrix. Now, the
vector of phenotypic observations y has a mean µ and a variance V, with the latter being determined by
the sum of the variances in the additive genetic effects, dominance genetic effects and the residuals,
i.e.
V = G1 + G2 + R
(Eq. S9)
The likelihood of the model in Eq. S2 can now be calculated from the probability density function for
the observations y under the assumption of normally distributed additive genetic effects, dominance
genetic effects and residual errors. Hence, in practice, given the observed data and the known family
relationships between observed individuals (derived from the mating design in Table 1 in the main
paper), the likelihood of the model is a function of µ and V, with the latter being a function of 𝜎𝑎2 , 𝜎𝑑2 ,
and 𝜎𝑒2 (see Eq. S4, S5, and S8). Restricted maximum likelihood estimation (REML) now maximizes
the likelihood of the model in Eq. S2 by determining REML estimates of µ, 𝜎𝑎2 , 𝜎𝑑2 , and 𝜎𝑒2 . In
practice, only y, A and D needed to be fed as input into the SAS 9.2 ‘PROC MIXED’ function for the
REML estimation to obtain these estimates. Finally, to discriminate the ‘true’ population parameters,
i.e. µ, 𝜎𝑎2 , 𝜎𝑑2 , and 𝜎𝑒2 , from their REML estimates, the estimates are denoted as µ
̂,VA,VD and VE,
respectively, in the main paper.
References
Lynch M, Walsh B (1998) Genetics and Analysis of Quantitative Traits. Sinauer Associates,
Sunderland, Massachusetts, USA.
Kruuk LEB (2004) Estimating genetic parameters in natural populations using the ‘animal
model’. Phil Trans. Roy. Soc. B Biol. Sci., 359: 873-890.
S2. Chemical composition of test media
Table S1: Physico-chemical characteristics of test media during the life table experiment. Values represent mean ±
standard deviation. NM is new medium. OM is old medium.
Nominal Cd
concentration
(µg Cd/L)
DOC (mg C/L)
pH
Cd concentration Mean
(µg Cd/L)
concentration
Cd
(µg Cd/L)
0
5
0
5
NM
4.68
OM
5.33 ± 0.42
NM
4.7
OM
5.62 ± 0.41
NM
4.62
OM
5.41 ± 0.51
NM
4.58
OM
5.41 ± 0.38
< D.L.
7.47 ± 0.13
< D.L.
4.52 ± 0.07
7.53 ± 0.17
0.00
< D.L.
4.56 ± 0.08
7.61 ± 0.15
3.64
2.76 ± 0.03
< D.L.
7.46 ± 0.17
0.00
3.81
3.05 ± 0.08
S3. Tables with all observed phenotypic values of net reproductive rate
Introductory note to Tables S2-S25:
These tables contain all observed net reproductive rate (R0) data for all individuals tested from
all clonal lineages, as obtained from the life-table experiments. Tables S2-S3, S8-S9, S14S15, and S20-S21 present the R0 of all individuals (‘replicates’) from the parental clonal
lineages. The column labels in these tables are single letters representing the identity (or clone
name) of the parental clonal lineage. All other tables present the R0 of the individuals from
the offspring clonal lineage. Here the column labels are ‘compound’ clone names consisting
of two letters, the first representing the corresponding maternal clonal lineage that was used in
the line crossing to produce the offspring lineage, the second representing the corresponding
paternal clonal lineage. Clone names are the same as those used in Table 1 of the main text. If
cells do not contain a value, this means a ‘missing’ observation, either because the individuals
set up appeared to be male instead of female or because they were accidentally lost (e.g. due
to manipulation error).
Table S2: Values of net reproductive rate at 20°C.
Clone 
A
B
C
D
F
G
H
J
K
N
93
117
61
116
113
137
68
126
72
92
127
33
132
115
105
96
120
104
78
111
161
24
105
108
67
110
152
78
Replicate ↓
1
2
3
92
Table S3: Values of net reproductive rate at 20°C.
Clone 
O
P
R
S
U
V
W
Y
E
1
136
88
135
141
149
48
211
124
125
2
27
37
122
179
97
182
142
136
58
75
121
143
I
Replicate ↓
3
58
Table S4: Values of net reproductive rate at 20°C.
Clone 
CW
EF
EY
RK
1
109
71
46
110
2
111
78
0
3
119
60
79
IH
NK
GV
108
RD
ID
FN
116
140
126
127
109
168
114
131
117
114
113
Replicate ↓
140
109
Table S5: Values of net reproductive rate at 20°C.
Clone 
SY
AW
DU
HU
YW
IS
1
139
84
134
117
25
114
2
0
111
135
113
0
3
96
112
99
98
95
EW
CR
RU
RJ
158
168
151
151
128
137
151
148
118
EN
VF
CE
29
125
166
Replicate ↓
80
114
Table S6: Values of net reproductive rate at 20°C.
Clone 
BA
OA
GW
KE
PV
JH
JY
1
84
66
135
60
116
2
61
135
104
52
87
145
14
104
113
3
34
0
127
67
101
118
2
112
92
Replicate ↓
Table S7: Values of net reproductive rate at 20°C.
Clone 
FY
FD
SP
HR
SD
NA
RN
DA
NU
121
134
119
146
132
96
129
117
127
124
152
133
118
115
145
118
43
131
189
121
Replicate ↓
1
2
3
102
97
Table S8: Values of net reproductive rate at 20°C and Cd.
Clone 
A
B
C
D
F
G
H
J
K
N
84
90
83
123
111
120
93
98
61
102
87
82
89
107
89
94
103
95
84
Replicate ↓
1
2
3
84
70
44
103
78
99
103
110
89
113
I
Table S9: Values of net reproductive rate at 20°C and Cd.
Clone 
O
P
R
S
U
V
W
Y
E
1
103
72
48
97
123
88
121
108
60
2
124
72
107
155
82
146
128
46
33
117
81
21
126
57
NK
GV
RD
ID
FN
93
89
0
61
67
72
0
0
95
54
83
126
66
EW
CR
RU
RJ
Replicate ↓
3
Table S10: Values of net reproductive rate at 20°C and Cd.
Clone 
CW
EF
EY
RK
1
100
87
0
2
74
98
42
3
105
85
49
IH
Replicate ↓
16
Table S11: Values of net reproductive rate at 20°C and Cd.
Clone 
SY
AW
DU
HU
YW
IS
1
125
81
100
0
116
77
90
120
88
2
0
116
102
123
68
93
111
87
3
115
91
87
94
4
106
92
85
EN
VF
CE
28
61
117
27
78
101
Replicate ↓
Table S12: Values of net reproductive rate at 20°C and Cd.
Clone 
BA
OA
GW
KE
PV
JH
1
52
97
68
90
113
2
55
92
103
82
83
JY
Replicate ↓
83
3
61
0
103
64
80
81
6
90
0
Table S13: Values of net reproductive rate at 20°C and Cd.
Clone 
FY
FD
SP
HR
SD
NA
RN
DA
NU
107
92
90
101
140
82
72
48
142
91
107
129
95
111
31
96
137
123
119
85
114
31
Replicate ↓
1
2
3
29
Table S14: Values of net reproductive rate at 24°C.
Clone 
A
B
C
D
F
G
H
J
K
N
1
144
109
76
144
136
110
156
120
148
159
2
140
103
91
171
24
111
135
109
138
161
3
16
152
179
114
101
154
114
161
151
I
Replicate ↓
Table S15: Values of net reproductive rate at 24°C.
Clone 
O
P
R
S
U
V
W
Y
E
1
183
109
30
129
105
112
148
136
120
2
175
98
86
131
90
92
115
169
138
3
84
95
43
124
156
80
164
125
Replicate ↓
Table S16: Values of net reproductive rate at 24°C.
Clone 
CW
EF
1
86
2
3
EY
RK
IH
NK
GV
RD
ID
FN
94
72
165
99
89
206
0
67
133
76
98
161
100
117
0
75
76
162
77
76
0
67
102
116
0
Replicate ↓
Table S17: Values of net reproductive rate at 24°C.
Clone 
SY
AW
DU
HU
YW
IS
EW
CR
RU
RJ
1
126
70
99
73
119
0
137
155
120
148
2
0
106
147
91
49
141
0
119
152
3
132
80
177
164
121
97
142
122
0
Replicate ↓
Table S18: Values of net reproductive rate at 24°C.
Clone 
BA
OA
GW
KE
PV
JH
JY
EN
VF
CE
1
23
6
154
141
72
0
93
0
98
21
2
63
0
91
75
106
137
162
5
97
81
75
51
103
115
33
66
Replicate ↓
3
42
Table S19: Values of net reproductive rate at 24°C.
Clone 
FY
FD
SP
HR
SD
NA
RN
DA
NU
1
144
98
135
213
147
35
47
126
72
2
125
102
93
93
127
36
102
142
86
3
135
110
148
107
102
148
148
22
Replicate ↓
Table S20: Values of net reproductive rate at 24°C and Cd.
Clone 
A
B
C
D
F
G
H
J
K
N
1
7
1
0
18
7
0
13
102
8
9
2
5
16
0
12
10
0
22
39
0
4
3
28
89
0
10
0
10
33
0
0
I
Replicate ↓
Table S21: Values of net reproductive rate at 24°C and Cd.
Clone 
O
P
R
S
U
V
W
Y
E
1
0
4
0
4
7
0
3
12
3
2
0
4
0
6
8
6
0
0
6
3
0
3
0
0
8
0
12
4
Replicate ↓
Table S22: Values of net reproductive rate at 24°C and Cd.
Clone 
CW
EF
1
0
2
3
EY
RK
IH
NK
GV
RD
ID
FN
0
0
0
6
7
8
0
28
2
5
5
0
0
6
38
43
19
0
0
14
0
14
1
3
0
Replicate ↓
Table S23: Values of net reproductive rate at 24°C and Cd.
Clone 
SY
AW
DU
HU
YW
IS
EW
CR
RU
RJ
1
0
0
7
0
10
0
8
13
35
7
2
0
48
14
0
0
6
10
31
45
3
2
78
29
5
0
4
4
5
55
Replicate ↓
Table S24: Values of net reproductive rate at 24°C and Cd.
Clone 
BA
OA
GW
KE
PV
JH
JY
EN
VF
CE
1
32
21
3
6
14
74
0
0
15
18
2
12
68
0
7
3
71
3
0
28
20
8
62
58
1
6
0
Replicate ↓
3
0
Table S25: Values of net reproductive rate at 24°C and Cd.
Clone 
FY
FD
SP
HR
SD
NA
RN
DA
NU
1
10
26
2
7
65
0
8
0
5
2
0
15
0
10
36
6
0
0
13
3
0
0
6
0
4
0
0
3
Replicate ↓