Evolution enhances the mutational
robustness and suppresses theemergence of new phenotype
M. Kikuchi (collaborator: T. Kaneko)
Cybermedia Center, Osaka UniversityIntroduction

Living systems have developed mutational
robustness in the course of evolution.
Mutational robustness can be examined
experimentally by several methods such as geneknock-out or artiﬁcial addition of generegulation.
But the evolution of mutational robustness is
not easy to investigate experimentally. Thus
numerical methods play important roles.
In this talk, we focus on the evolution of mutational
robustness of a toy model of gene regulatorynetworks (GRN).

Evolutionary simulations provide
evolutional-path-dependent traits.
In order to study the universality and theparticularity of evolution, we need referencesystems.
The reference systems that we consider natural
are GRNs obtained by random sampling.
The preceding work
S. Nagata and M. Kikuchi ”Emergence ofcooperative bistability and robustness of gene
regulatory networks” PLoS Comput Biol 16(2020) e1007969

Gene Regulatory Network

Gene expression

Gene regulation bytranscription factors

Model and Methods

ModelWe ignore detailsof gene regulation.
We consider
neural-network-like
connectionist
model.
Node: Gene
Edge:Regulation

Network model
Directed random graphN = 32 nodes、K = 80 edges

One input gene and one output gene
All the possible networks are allowed

Discrete-time dynamics (Neural-network like)∑xi (t + 1) = R I δi,1 +Jij xj (t)jR(x) =1
1 + e −β(x−µ)

xi : Expression of ith gene ( [0, 1])
Jij : Regulation of ith gene by jth gene(0, ±1)+1: activation, −1: repression

I : Input from exterior world ([0, 1])µ = 0, β = 2

Fitness
We require that the expression of the output
node for I = 0 and 1 diﬀer as large as possible.
We deﬁne ﬁtness
f = |x̄out (0) − x̄out (1)|,where x̄out (I ) is the ﬁxed-point value (or thelong-time average if no ﬁxed point is reached)of the output node for input I .
f ∈ [0, 1] by deﬁnition

Methods
1

Random sampling by multicatnonical ensemble
methodIt enables us to sample GRNs in wide range ofﬁtness randomly (in principle).
Fitness are divided into 100 bins. The weight foreach bin is determined by Wang-Landau method.
5M samples

2

Evolutionary simulation
Mutation: random reconnection of a randomly
selected edge
1000 GRNs; 500 are selected at each generation
The lineage of the highest ﬁtness at 150 generation
is sampled
10000 independent runs

Results

Fitness landscape and evolutionary speed
0

Ω(f)

4060
random sampling
evolution

0.0

0.2

80

0.4

0.6

0.8

1.0

100

f

100

0

10−1

20

Generation

Generation

20

10−2

40

Ω(f)

10−1
10−3
10−5
10−7
10−9
10−11
10−13
10−15

10−36010−4

random samplingevolution

80

10−5100
0.0

0.2

0.4

0.60.81.0

fProbability of ﬁtness
and ﬁtness vs.
generation for β = 2for β = 4 (only thisﬁgure)1234

Most GRNs have low ﬁtness (f < 0.1).
At intermediate ﬁtness, GRNs become
exponentially rare as ﬁtness increases.
For very high ﬁtness (f > f ∗ ), GRNs become
faster-than-exponentially rare.
Evolution slows down suddenly at f ≃ f ∗ .

The evolutionary speed is determined mainly by thenumber of available GRNs (or entropy).

Emergence of bistability1.0

random sampling
evolution

1.0up

0.8

down

P2 ( f )0.8x(I)

0.6

0.4

0.6
0.4
0.2

0.2

0.0
0.2

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.3

0.4

0.5

0.6

f

0.7

0.8

0.9

1.0

I

Example of bistability
for f = 0.9

The fraction of thebistable GRNs vs.
ﬁtness1

Universality of evolution:
Bistable GRNs emerge, and their fraction grows
as ﬁtness increases. All GRNs become bistableas f → 1.
Bistability is a new phenotype. This newphenotype that was not explicitly required by theﬁtness appears inevitably.

2Particularity of evolution:
Evolution delays appearance and growth in thefraction of the bistable GRNs.
Evolution is conservative!

Mutational robustness
Robustness measure
We consider the single-edge deletion asmutation.
Delete a single edge from GRN and computeﬁtness f ′ of mutated GRN
Robustness measure for a given GRN is theaverage of f ′ for all the possible single-edge
deletion.
1 ∑ ′r≡fiKifi ′ is f ′ for GRN that ith edge is deleted.

We compute the average ⟨r ⟩ over all GRNs ineach bin.
Since ⟨r ⟩ increases with f , it is not suitable tocompare mutational robustness of diﬀerent
ﬁtness.
We focus on diﬀerence in ⟨r ⟩ of GRNs having
the same f obtained by random sampling andevolution.

1.0

r⟩ndom⟨s⟩mpling
evolution

0.8

⟨r⟩

0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

fRobustness measure vs. ﬁtness

Probability distribution of r
0.200

random sampling
evolution

0.35
0.30

0.150

0.25

0.125

0.20

P( r )P( r )

random sampling
evolution

0.175

0.15

0.100
0.075

0.10

0.050

0.05

0.025
0.0000.00
0.0

0.2

0.4

r

0.6

f = 0.5

0.8

1.0

0.0

0.2

0.4

r

0.6

f = 0.8

0.8

1.0Probability distribution of r
0.10

random sampling
evolutionP( r )

0.08
0.06
0.04
0.02
0.00
0.0

0.2

0.4

r

f = 0.99

0.6

0.8

1.0

Two-step evolution
1
In the early stage of evolution, ⟨r ⟩ coincides
with random sampling.
2
In the late stage of evolution, ⟨r ⟩ exceeds
random sampling.
Particularity of evolution:
The evolutionally obtained probability
distribution of r for high ﬁtness is signiﬁcantlybiased to the robust side compared to random
sampling.
Evolution enhances the mutational robustness!

Explore the origin of robustness
0.5

0.4

0.06

P(nL)

P(f')

random sampling
evolution

0.08

0.3

0.04

0.2

0.02
0.1

0.00
0.0

0.0

0.2

0.4

f'

0.6

0.8

1.0

Distribution of f ′ for
f = 0.99010203040

nL

50

60

7080

Distribution of lethaledges for f = 0.99

0.200

random network
random sampling
evolution

0.175
0.150

P(nN)

0.125
0.100
0.075
0.050
0.025
0.000
051015nN

202530Distribution of eﬀective network size for f = 0.99

For high ﬁtness, edges are divided in twoclasses: neutral and lethal
The number of lethal edges are signiﬁcantly
scarce in evolutionally obtained GRNs.
The origin of the diﬀerence in the mutational
robustness is the diﬀerence in the number of lethaledges.

The eﬀective network size does not aﬀect themutational robustness.

Network motifs
Deﬁnition of motifs
The connection patterns appear in the given
network at frequencies much higher than thosefound in random networks.
We explore the following patterns for f = 0.99
12345

auto-regulation
mutual regulation
feedforward loop
feedback loop
mutual activation/repression plus auto-activation

random_network
random_sampling
evolution

0.35
0.30

P(nS + )

0.25
0.20
0.15
0.10
0.05
0.00
0246

nS +

810

Auto-activation

12

0.5
0.4
0.3
0.2

random network
random sampling
evolution

0.4

P(nm − )P(nm + )0.5

random network
random sampling
evolution

0.3
0.2
0.1

0.1

0.0

0.0
0246

nm +

810

Mutual activation

120246

nm −810Mutual repression

12random network
random sampling
evolution

0.175

0.30

random network
random sampling
evolution

0.25

0.125
0.100

P(nFBL + )P(nFFL + )

0.150

0.075
0.050

0.20
0.15
0.10

0.025

0.05

0.000
0510

nFFL +

15

20

0.00
02

46

nFBL +

810

12

Coherent feed-forward
Positive feedback loop
loop

1.0

random network
random sampling
evolution

P(nMA − )

0.8
0.6
0.4
0.2
0.0
012

nMA −

345

mutual repression plus
auto-activation

Six patterns may be called as motif (but not sosigniﬁcant)+FFL is ubiquitous motif in GRNs
Other patterns are known as motifs in multistable
GRNs.

Motif distributions do not diﬀer for
evolutionally obtained GRNs and randomly
sampled GRNs.
Motifs are related to the function but not to themutational robustness.
Global structures are considered to be relevant tothe mutational robustness.

Path distribution
The number of paths connecting input and output
nodes
80

random sampling
evolution

70
60
random sampling
evolutionP(npath)

0.005

50

nL0.006

400.004

300.00320
100.00200.00100.000−0.001050

100

150200

npath

250

300

350

400

200

400

npath

600

800

1000

The number of paths
vs. the number oflethal edges

12The probability distribution of the number ofpaths npath connecting input and output nodes
are signiﬁcantly diﬀerent for a small number ofpaths, npath = 1 in particular.
This diﬀerence does not fully explain themechanism for enhancement of the mutational
robustness by the evolution, because the lethaledges are signiﬁcantly scarce irrespective of thenumber of paths compared torandomly-produced GRNs.Discussion

Mechanism
Mutation consists of two successive processes:
12

Deletion of a randomly selected edge
Addition of a new edge

When a lethal edge is deleted in the ﬁrst
process, the ﬁtness of the GRN drops down toalmost zero, and the possibility is very low thatthe ﬁtness recovers by the second process.
GRNs with many lethal edges are more likely tobe eliminated than those with fewer lethaledges.
Second-order selection (Wagner)

Summary

Method
We compared the evolution of GRN withrandom sampling achieved by themulticanonical ensemble.
Results1Evolution delays the emergence of a newphenotype.
Evolution is conservative.
2

Mutational robustness evolves.
GRNs with fewer lethal edges are selected byevolution.
Motifs are not relevant to mutational robustness.
The number of paths has some relevance tomutational robustness.

The results are intuitively natural. But we were able
to discuss them quantitatively by the combinationof the Multicanonical Monte Carlo method and theevolutionary simulation.

Selected references
Robustness
A. Wagner ”Robustness and Evolvability in Living
Systems” (2005) Princeton

MethodY. Iba, N. Saito N and A.Kitajima ”Multicanonical
MCMC for sampling rare events: an illustrative
review” Ann Inst Stat Math. 66 (2014) 611

MotifsU. Alon ”An Introduction to Systems Biology (second edition)” (2020) CRC Press