Supplementary Information for How development affects evolution

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Supplementary information for:
How development affects evolution
Mauricio González-Forero*1
1School of Biology, University of St Andrews, Dyers Brae, St Andrews, KY16 9TH, Fife, UK
Contents
S1 Matrix calculus notation 1
S2 Exploring adaptation without selection 1
S3 Development enables negative senescence 3
S1 Matrix calculus notation
Following Caswell (2019), we use the following notation from matrix calculus. The Jacobian matrix of a vector a∈Rn×1
with respect to a vector b∈Rm×1in its standard or transposed form is, respectively,
∂a
∂bÖ=






∂a1
∂b1
··· ∂a1
∂bm
.
.
.....
.
.
∂an
∂b1
··· ∂an
∂bm







∈Rn×mor ∂aÖ
∂b=






∂a1
∂b1
··· ∂an
∂b1
.
.
.....
.
.
∂a1
∂bm
··· ∂an
∂bm







∈Rm×n. (S1)
The transpose of ∂a/∂bÖis (∂a/∂bÖ)Ö=∂aÖ/∂b. The analogous notation applies for total derivatives.
S2 Exploring adaptation without selection
A debated question is whether adaptation may arise from plasticity without selection (West-Eberhard, 2003; Laland
et al., 2015). This question may be addressed using Eq. (19), whereby the evolutionary dynamics of developed pheno-
types consists of selection response and exogenous plastic response. Here we ask if exogenous plastic response alone
can lead to adaptation in a best-case scenario for adaptive plasticity, where development is selective.
From Eq. (19) in the main text, we have that exogenous plastic response of the geno-phenotype is
µsz
s²
²
²Ö
∂²
²
²
∂τ ¶¯¯¯¯y=¯
y
.
More specifically, from Layer 7, Eq. 1a of González-Forero (2021), we have that the exogenous plastic response of the
phenotype is
µsx
s²
²
²Ö
∂²
²
²
∂τ ¶¯¯¯¯y=¯
y
.
To make the exogenous plastic response as adaptive as possible, we seek to write the stabilized plasticity of the
phenotype sx/s²
²
²Ö|y=¯
yin terms of selection gradients, by letting development be selective. Using Layer 5, Eqs. 2b and
1; Layer 4, Eqs. 5 and 4; and Layer 3, Eq. 4 of González-Forero (2021), we have that the exogenous plastic response of
the phenotype is
sx
s²
²
²Ö
∂²
²
²
∂τ =sx
s¯
xÖ
dx
d²
²
²Ö
∂²
²
²
∂τ =µI−dx
d¯
xÖ¶−1dx
d²
²
²Ö
∂²
²
²
∂τ =µI−dx
dxÖ
δx
δ¯
xÖ¶−1dx
dxÖ
∂x
∂²
²
²Ö
∂²
²
²
∂τ .
*Corresponding author: mgf3@st-andrews.ac.uk
1

The two rightmost matrices are direct plasticity and exogenous environmental change, which give the vector of direct
exogenous plastic response:
∂x
∂²
²
²Ö
∂²
²
²
∂τ =ÃNa
X
j=1
∂xa
∂²
²
²Ö
j
∂²
²
²j
∂τ !=Ã∂xa
∂²
²
²Ö
a−1
∂²
²
²a−1
∂τ !=Ã∂ga−1
∂²
²
²Ö
a−1
∂²
²
²a−1
∂τ !=ÃNe
X
k=1
∂gi,a−1
∂²k,a−1
∂²k,a−1
∂τ !,
where the second equality follows from Layer 2, Eq. 2c of González-Forero (2021). The last expression is in terms of
age-specific direct plasticity ∂gi,a−1/∂²k,a−1, which we now make a function of the selection gradient of the corre-
sponding phenotype. Thus, we let development be selective. Specifically, let
∂gi,a−1
∂²k,a−1
=Vik,a,a−1
∂w
∂xi,a−1
, (S2)
for a∈{2, . .. , Na} and some differentiable function Vik,a,a−1(ma−1,¯
z), where the selection gradient ∂w/∂xi,a−1is a
function of ma−1and ¯
zfrom Eq. (4) of the main text. Consequently, our assumptions for the developmental con-
straint (1) of the main text are met where the developmental map gais a function of mutant traits at the same age.
It would be convenient that ∂gi,a−1/∂²k,a−1were also a function of ∂²k,a−1/∂τ to cancel exogenous environmental
change and keep only selection gradients, but this is not allowed because this violates our assumption in the de-
velopmental constraint (1) of the main text that gadepends on the environment at that age but not on the rate of
exogenous environmental change: if we were to let the developmental map depend on the rate of exogenous environ-
mental change, there would be a third term in the equation of the evolutionary dynamics of the phenotype, a term
that would depend on the exogenous environmental acceleration over evolutionary time.
Now, let us define the matrix
Va+1,a=


V11,a+1,a· · · V1Ne,a+1,a
.
.
.....
.
.
VNp1,a+1,a· · · VNpNe,a+1,a


∈RNp×Ne,
for a∈{1, . .. , Na−1}. Then, using Eq. (S2), we can write the matrix of age-specific direct plasticity for age a∈{2, . .. , Na}
as
∂xa
∂²
²
²Ö
a−1
=








∂w
∂x1,a−1
V11,a+1,a· · · ∂w
∂x1,a−1
V1Ne,a+1,a
.
.
.....
.
.
∂w
∂xNp,a−1
VNp1,a+1,a· · · ∂w
∂xNp,a−1
VNpNe,a+1,a








=












∂w
∂x1,a−1
0··· 0
0∂w
∂x2,a−1
··· 0
.
.
..
.
.....
.
.
0 0 ··· ∂w
∂xNp,a−1















V11,a+1,a· · · V1Ne,a+1,a
.
.
.....
.
.
VNp1,a+1,a· · · VNpNe,a+1,a



=diagµ∂w
∂xa−1¶Va,a−1,
where diag(x) is the diagonal matrix with vector xin its main diagonal. Hence, we can write the matrix of direct
plasticity as
∂x
∂²
²
²Ö=UV,
where we define a matrix involving selection gradients as
U=









0 0 ··· 0
0diagµ∂w
∂x1¶··· 0
.
.
..
.
.....
.
.
0 0 ··· diagµ∂w
∂xNa−1¶









=diagµµ0;diagµ∂w
∂x1¶;··· ;diagµ∂w
∂xNa−1¶¶¶
2

and another matrix with the unspecified functions as
V=








0 0 ··· 0 0
V21 0··· 0 0
0 V32 ··· 0 0
.
.
..
.
.....
.
..
.
.
0 0 ··· VNa,Na−10








.
Then, with selective development of the form (S2), the vector of direct exogenous plastic response is of the form
∂x
∂²
²
²Ö
∂²
²
²
∂τ =UV ∂²
²
²
∂τ =ÃNa
X
j=1
∂xa
∂²
²
²Ö
j
∂²
²
²j
∂τ !=Ã∂xa
∂²
²
²Ö
a−1
∂²
²
²a−1
∂τ !=µdiagµ∂w
∂xa−1¶Va,a−1
∂²
²
²a−1
∂τ ¶.
Thus, the best-case scenario for adaptation via exogenous plastic response given by Eq. (S2) yields a matrix Uthat
involves the selection gradient with a lag in developmental time (the a-th block entry of the vector of direct exoge-
nous plastic response depends on a diagonal matrix of the direct selection gradients of the phenotype at age a−1).
Consequently, if there is an optimum phenotypic value at each age and it changes substantially with age, selective
development of the form (S2) may yield maladaptive exogenous plastic response. Moreover, adaptation via exoge-
nous plastic response with selective development of the form (S2) is prevented if the rate of change in exogenous
environmental change changes in sign over evolutionary time (i.e., if ∂²ka /∂τ changes sign).
S3 Development enables negative senescence
Here we derive Eq. (20) of the main text. From Eq. (19) in the main text, we have that evolutionary change in the
geno-phenotype due to natural selection is given by
ιLzy
dw
dy¯¯¯¯y=¯
y
, (S3a)
where the mechanistic socio-genetic cross-covariance matrix between the geno-phenotype and genotype is
Lzy =sz
syÖHy=µLxy
Hy¶(S3b)
and
Hy=cov[y,y] (S3c)
is equivalently the mutational covariance matrix (of the genotype) and the mechanistic additive genetic covariance
matrix of the genotype. The matrix sz/syÖis a “stabilized” total derivative and it is non-singular because of our as-
sumption that the genotype is developmentally independent (Appendix H of González-Forero, 2021). Hence, Lzy is
non-singular if Hyis non-singular. Therefore, if ι6= 0 and Hyis non-singular, then selection on the geno-phenotype
vanishes if and only if dw/dy=0|y=¯
y.
The total selection gradient of the genotype is
dw
dy¯¯¯¯y=¯
y
=µδw
δy+dxÖ
dy
δw
δx¶¯¯¯¯y=¯
y
(S4)
(Layer 4 Eq. 22 of González-Forero 2021).
Eq. (4) in the main text gives a mutant’s relative fitness in terms of generation time, which is
T=
Na
X
j=1
j`◦
jf◦
j, (S5a)
(Eq. 6 of González-Forero 2021; Charlesworth 1994, Eq. 1.47c; Bulmer 1994, Eq. 25, Ch. 25; Bienvenu and Legendre
2015, Eqs. 5 and 12). The superscript ◦denotes evaluation at y=¯
y, so f◦
jand p◦
jare, respectively, the fertility and
survival probability of a neutral mutant at age j. A mutant’s relative fitness (Eq. 4) also depends on the forces of
selection on fertility and survival, which are respectively
φj=`◦
j(S5b)
πj=1
p◦
j
Na
X
k=j+1
`◦
kf◦
k, (S5c)
3

(Eqs. 7 of González-Forero 2021; Hamilton 1966 and Caswell 1978, his Eqs. 11 and 12), where the survivorship of
neutral mutants is `◦
j=Qj−1
i=1p◦
i.
From Eqs. (S3), if the genotypic traits are mutationally uncorrelated (i.e., Hyis diagonal), the evolutionary change
of the i-th genotypic trait at age a,¯
yi a , due to selection is given by
ιHyi a
dw
dyi a ¯¯¯¯y=¯
y
. (S6)
Since Hyia is a variance, it is non-negative, so the i-th resident genotypic trait at age aincreases over evolutionary time
if and only if the total selection gradient of this genotypic trait is positive, provided that there is mutational variation
for this genotypic trait at that age (i.e., ιHyia >0). From (S4), the total selection gradient of the genotypic trait yi a is
dw
dyi a ¯¯¯¯y=¯
y
=Ãδw
δyi a
+
Np
X
k=1
Na
X
j=1
dxk j
dyi a
δw
δxk j !¯¯¯¯¯y=¯
y
=Ãδwa
δyi a
+
Np
X
k=1
Na
X
j=1
dxk j
dyi a
δwj
δxk j !¯¯¯¯¯y=¯
y
, (S7)
where the second equality follows because total immediate derivatives do not consider developmental constraints.
Let us now see that the forces of selection decrease, or remain constant, with age as has been long established
(Hamilton, 1966; Wensink et al., 2017; Caswell and Shyu, 2017). The change in the force on fertility with age is
φj+1−φj=`◦
j+1−`◦
j=`◦
jp◦
j−`◦
j=`◦
j(p◦
j−1) ≤0, (S8a)
where the rightmost inequality follows because p◦
jis a probability. Hence, the force on fertility is non-increasing with
age. In turn, the change in the force on survival with age is
πj+1−πj=1
p◦
j+1
Na
X
k=j+2
`◦
kf◦
k−1
p◦
j
Na
X
k=j+1
`◦
kf◦
k
=Ã1
p◦
j+1
−1
p◦
j!Na
X
k=j+2
`◦
kf◦
k−1
p◦
j
`◦
j+1f◦
j+1, (S8b)
which is non-positive if p◦
j→p◦
j+1as is the case if the survival probability changes smoothly with age.
Now, suppose that the genotypic trait yi a has a deleterious effect on survival or fertility at an early age asuch that
δwa/δyi a <0, and a pleiotropic, beneficial effect on survival or fertility of similar magnitude at a later age j>asuch
that δwj/δxk j >0 for some phenotype xk j , but no other fitness effects. Then, total immediate selection is weaker at
the later age because of declining selection forces (i.e., |δwa/δyia | ≥ |δwj/δxk j |). Yet, from (S7) we obtain that such
genotypic trait is favoured if its total effect on the phenotype is sufficiently large, such that
µδwa
δyi a
+dxk j
dyi a
δwj
δxk j ¶¯¯¯¯y=¯
y
>0.
In such case, from (S6), the resident genotype ¯
yi a increases if its mutation rate and mutational variance are non-zero.
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4

Directional
selection
on genotype
Directional
selection
on phenotype
Developmental
bias from
genotype
Developmental
bias from
phenotype
Niche construction
by phenotype
Niche construction
by genotype
Phenotypic
plasticity
Environmental
sensitivity
of selection
Total effects of
genotype on phenotype
Total genotypic
selection
Total immediate genotypic
selection
Total immediate phenotypic
selection
Total effects of
phenotype on phenotype
Total immediate effects of
genotype on phenotype
Total immediate effects
of phenotype on phenotype
Figure S1: Total genotypic selection depends on many factors. Total genotypic selection — which is measured by
the total selection gradient of the genotype dw/dy— describes selection response relatively completely in that it is
premultiplied by a non-singular matrix Lzy if there are no absolute mutational constraints, so total genotypic selection
can identify evolutionary equilibria. In contrast, direct phenotypic and genotypic selection — measured by ∂w/∂z—
and total phenotypic and genotypic selection — measured by dw/dz— describe selection response less completely in
that they are always premultiplied by a singular matrix and so do not generally identify evolutionary equilibria. Total
genotypic selection depends on direct directional selection, developmental bias, plasticity, niche construction, and
environmental sensitivity of selection. An arrow from a variable to another one indicates that the latter depends on
the former.
Laland, K.N., Uller, T., Feldman, M.W., Sterelny, K., Müller, G.B., Moczek, A. et al (2015). The extended evolutionary
synthesis: its structure, assumptions and predictions. Proc. R. Soc. B,282, 20151019.
Wensink, M.J., Caswell, H. and Baudisch, A. (2017). The rarity of survival to old age does not drive the evolution of
senescence. Evol. Biol.,44(1), 5–10.
West-Eberhard, M.J. (2003). Developmental Plasticity and Evolution. Oxford Univ. Press, Oxford, UK.
5