Phenotype Selection Due To Mutational Robustness: A Computational Study
Macoto Kikuchi (Cybermedia Center, Osaka University, Japan)

AbstractModel

Essential edge distribution for f = 0.9

The mutation-selection mechanism of evolution induces the enhancementof mutational robustness. Suppose that more than one phenotypes havethe same ﬁtness. We expect that some phenotypes are more likely to be
selected while some are less likely to be selected as a result of thedifference in mutational robustness, even if they have the same ﬁtness.
We studied this phenotype selection for a model of gene regulatory
networks (GRNs). We generated a random reference ensemble using themulticanonical Monte Carlo (McMC) method and compared it withevolutionary simulations (Evo) for investigating particular properties ofevolution. By comparing the ratio of the toggle switches and the one-way
switches between McMC and Evo for one-in-one-out GRNs, wedemonstrate that the one-way switches are suppressed in evolution due tomutational robustness.

Connectionist-type model GRNs
are expressed by directed
graph, ignoring details ofgene expression.
Node→Gene,Edge→Regulatory
relations.
40 nodes, 120 edges
One input and one output

Essential edges: Edges that the ﬁtness drops largely when cut.

xi (t + 1) = R Iδi,0 +1

Jij xj (t) , R(y) =Mutation: single-edge rewiring.
Initial population: 1000 random GRNs.
500 GRNs are selected at each generation based on ﬁtness.
The lineage of the highest ﬁtness at ﬁnal generation is sampled.
Distribution over many lineages.
2

Multicanonical MC (McMC)
A set of random GRNs are generated by McMC.
Random GRNs in the whole range of ﬁtness are obtained.
Fitness is divided into 100 bins.
Weight for McMC is determined by Wang-Landau method

0.00

0.00

1020

0.2

0.6

xout

xout

xout

0.4

0.4

0.2

0.4

I

0.6

0.8

Monostable

1.0

0.0

0.2

2030

40

Essential Edges

50

60

0.000

0.00

1020

30

40

Essential Edges

0.4

I

0.6

0.8

P

0.6

0.002

0.4

0.000

1.0

0.0

Toggle switch

0.2

0.4

I

50

60

0.6

0.8

1.0

010

2030

40

Essential Edges

5060

Toggle switch
McMC
Evo

0.006
0.004

up
down

0.04
0.02

0McMC
Evo

0.06

0.005

0.80.20.2
0.0

10

0.08

0.008

1.0up
down

0.80Evolution

0.010

0.012

By changing I gradually, three types of stabilities are observed.
up
down60

McMC
Evo

0.010

1.050

Monostable

Classiﬁcation of stability

0.6

40

0.015

where xout (I) : Expression level of the output gene for input I.

0.830

Essential Edges

f = xout (1) − xout (0),010203040

Essential Edges

5060

In case of one-way switches, GRNs
with large number of essentialedges, namely, mutationally lessrobust GRNs, are largely suppressed.

One-way switch

One-way switch

Conclusion
Stability vs. ﬁtness

One-way switches are mutationally less robust and are suppressed inevolution.
2 This is an example of phenotype selection due to mutationalrobustness.
3 McMC is useful to reveal characteristic properties of evolution.
1

1.0

Bistable GRNs (toggleswitches and one-wayswitches) appear for highﬁtness.
2 Emergence of bistable GRNs
”delays” in Evo compared with
McMC.
10.8

McMC: One-way
Evo: One-way
McMC: Mono
Evo: Mono
McMC: Toggle
Evo: Toggle

0.6

P

Evolutionary simulation (Evo)

0.02

Essential edge distributions obtained by McMC and Evo for f = 0.9

xi : Expression level of i-th gene ([0, 1])
Jij : Regulation from j-th gene to i-th gene (0, ±1)+1: activation, −1: repression
I : Input signal ([0, 1])
β = 2, µ = 0.424: Spontaneous expression is 0.3
Fitness Obtain the steady state for I = 0 → Change suddenly to I = 1 andobtain the steady state.

Method
10.01

Essential edge distribution for each stability

1 + e−β(y −µ)

Introduction
Background: Selection by mutational robustness
Because of the mutation-selection mechanism of evolution, there is aselection bias towards mutationally robust genotypes. Since this bias
is not directly related to function, the selection due to the mutationalrobustness is called ”second order”
2 Proposal: Phenotype selection due to the second-order selection
Suppose more than one phenotypes have the same ﬁtness. Since thedistribution of mutational robustness should differ for different
phenotypes, we expect that some phenotypes are suppressed
because of the selection due to the mutational robustness.
3 Aim of the research
We investigate the above phenotype selection for a model of gene
regulatory networks (GRNs).
4 Methodology: Use of multicanonical MC
To investigate characteristic properties of evolution, we need areference system to compare with evolutionary simulation (Evo). The
appropriate reference is a set of randomly generated GRNs. For that
purpose, we use the multicanonical MC (McMC).

0.04

McMC (random reference set)

j1

PP
∑

0.02

0

Mono
ToggleOne-way

0.06

P

0.03

P

Discrete-time dynamics

0.04

Example of a small GRN

0.08

MonoToggle
One-way

0.4
0.2
0.0

0.5

0.6

0.7f

0.8

0.9

1.0

references
N.Saito and M.Kikuchi (2013) New J. Phys. 15, 053037.
S.Nagata and M.Kikuchi (2020) PLoS Comput Biol 16, e1007969.
T.Kaneko and M.Kikuchi (2022) PLoS Comput Biol 18, e1009796.