Phenotypic Noise and the Cost of Complexity

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Phenotypic Noise and the Cost of Complexity

Charles Rocabert, Guillaume Beslon, Carole Knibbe, Samuel Bernard

To cite this version:

w(z) is convex w(z) is concave w(z) is convex Inflection points Log-fitness plateaus 0.00 0.25 0.50 0.75 1.00 -2 0 2 Phenotype z Ab so lu te fi tn ess w (z)

Generalized fitness function: w(z) = (1-β)exp⎡⎣ − αzQ⎤

⎦ + β 0 5 10 15 20 25 2.5 5.0 7.5 10.0 Phenotypic complexity (n) R el at ive co nve rg en ce ti me

Gaussian-shaped fitness function

A -1.00 -0.75 -0.50 -0.25 0.00 2.5 5.0 7.5 10.0 Phenotypic complexity (n) R el at ive co nve rg en ce ti me

Rescaled fitness function

B Phenotypic noise mutation rate 1) Lower 2) Equal 3) Higher 4) No noise

Phenotypic Noise and

the Cost of Complexity

Charles Rocabert

1,2

_{, Guillaume Beslon}

1,3

_,

Carole Knibbe

1,4

_{, Samuel Bernard}

1,5

1. Inria, France 2. Synthe2c and Systems Biology Unit, BRC, Szeged, Hungary 3. CNRS UMR5205 LIRIS, INSA Lyon, France 4. INSERM U1060 CarMeN, INSA Lyon, France 5. CNRS UMR5208 Ins2tut Camille Jordan, Univ Lyon 1, France This work was supported by the European Commission 7th Framework Programmes 659 (FP7-ICT-2013.9.6 FET ProacFve: Evolving Living Technologies), and EvoEvo Project 660 (ICT-610427)

• B

ACKGROUND

Theory predicts that phenotypic noise is posi2vely selected under direc9onal selec9on because it increases the mean fitness value, and counter-selected under stabilizing selec9on (1).

It has been suggested that under direc2onal selec2on, the fitness gain provided by phenotypic noise also promotes adap9ve evolu9on (2), while the link is unclear. Evolu2on on mul2ple phenotypic characters suffers from the cost of complexity (3). The impact of phenotypic noise in mul2dimensional phenotypes is less understood.

• M

ETHODS

We used a quan2ta2ve model to study the adap2ve evolu2on of organisms with mul2ple phenotypic traits under selec2on and evolvable phenotypic noise (4) in a generalized fitness landscape.

• R

ESULTS

Phenotypic noise promotes adapta2ve evolu2on under direc2onal and/or stabilizing selec2on if the logarithmic fitness plateaus. For mul2ple phenotypic characters under selec2on, the phenotypic noise evolves to a one- dimensional noise aligned with the direc2on of the fitness op2mum.

• C

ONCLUSION

Phenotypic noise can decrease the cost of complexity and promotes adap9ve evolu9on in flat regions of the fitness landscape.

1. P

HENOTYPIC

NOISE

EFFECT

ON

FITNESS

DEPENDS

ON

THE

SHAPE

OF

THE

FITNESS

LANDCAPE

(

SINGLE

CHARACTER

)

3. P

HENOTYPIC

NOISE

DIMENSIONALITY

REDUCTION

4. P

HENOTYPIC

NOISE

PROMOTES

ADAPTIVE

EVOLUTION

AND

DECREASES

THE

COST

OF

COMPLEXITY

0.0 0.5 1.0 1.5 2.0 0 1 2 3 4 Mean phenotype µ Ma xi ma l σ Mean population trajectory A 0.8 0.9 1.0 0 500 1000 1500 Generations Ma xi ma l σ co nt rib ut io n Phenotypic noise flattening B 0.80 0.85 0.90 0.95 1.00 0 500 1000 1500 Generations Ma xi ma l σ co rre la tio n

Phenotypic noise alignment with the optimum

C Phenotypic noise shape D 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Phenotypic character 1 Ph en ot yp ic ch ara ct er 2

• Phenotypic noise: variability in phenotypes of isogenic organisms in constant environment, aka developmental noise, phenotypic heterogeneity, cellular noise, biological noise, intra-genotypic variability, ...

• Direc9onal selec9on: selec2on far from fitness op2mum characterized by a convex (posi2ve second deriva2ve) fitness landscape.

• Stabilizing selec9on: selec2on close to fitness op2mum characterized by a concave fitness landscape. • Adap9ve evolu9on: capacity of increasing mean popula2on fitness as measured by the rate of increase of the log-fitness with respect to the mean phenotype. • Cost of complexity: Reduc2on of the frac2on of beneficial muta2ons when the number of phenotypic characters under selec2on increases. Selected references: (1) Lande (1980); Pal (1998); Kawecki (2000); Paenke et al (2007) ; Zhang et al (2009) Mol Sys Biol 5:299; Bruijning et al (2019) (2) Bódi et al (2017) PLOS Biol 15:e2000644; Duveau et al (2018) Elife 7:e37272 (3) Orr (2000) Evolu2on 54:13-20; Mar2n & Lenormand (2006) Evolu2on 60:893-907 (4) Ito et al (2009) Mol Sys Biol 5:264; Pelabon et al (2010) Evolu2on 64:1912-1925; Viñuelas et al (2012) Prog Biophys Mol Biol 110:44-53; Shen et al (2012) PLOS Gene2cs 8:e1002839; Metzger et al (2015) Nature 521:344-347 (5) Zhang et al (2009) Mol Sys Biol 5:299; Draghi et al (2019); Parameters Q = 2 Q > 2 β = 0 β > 0

Shape of the absolute fitness function concave (curvature < 0) concave (curvature < 0) convex (curvature > 0) convex (curvature > 0) Shape of the log-fitness

function does not plateau plateaus does not plateau plateaus

Does noise increase

mean absolute fitness? No No Yes Yes

Does noise promote

evolution? No Yes No Yes

Stabilizing selection

(close to the fitness optimum) (far from the fitness optimum)Directional selection

β > 0 Q > 2

2. A

MODEL

FOR

PHENOTYPIC

NOISE

EVOLUTION

zopt Ph en ot yp ic ch ara ct er 2 µ = (µ1, µ2) σ1 σ2

Evolvable phenotypic noise for two phenotypic characters

Phenotypic character 1 θ1 Mul9-dimensional phenotypic noise is built from the mutable genotype {μ, σ, 𝚹}: • Mean phenotype μ (mutable), • Variances σ2 _(mutable), • Rota2on angles 𝚹 (mutable), • Covariance Σ built from σ2 _{and 𝚹.} • Phenotype z ∼ 𝘕n(μ, Σ) For mul2ple phenotypic characters under direc2onal selec2on, we demonstrate that the best phenotypic noise configura2on is aligned and fully correlated with the direc9on of the fitness op9mum. Example: Simula2on for two phenotypic characters: • Ini2al distance: 4 units, • Popula2on size: 1,000, • Muta2on rate: 1e_-3, • muta2on size: 0.01. 0 5 10 15 20 25 2.5 5.0 7.5 10.0 Phenotypic complexity (n) R el at ive co nve rg en ce ti me

Gaussian-shaped fitness function

A

-1.00 -0.75 -0.50 -0.25 0.00 2.5 5.0 7.5 10.0 Phenotypic complexity (n) R el at ive co nve rg en ce ti me

Rescaled fitness function

B

Phenotypic noise mutation rate 1) Lower 2) Equal 3) Higher 4) No noise 0 5 10 15 20 25 2.5 5.0 7.5 10.0 Phenotypic complexity (n) R el at ive co nve rg en ce ti me

Gaussian-shaped fitness function A -1.00 -0.75 -0.50 -0.25 0.00 2.5 5.0 7.5 10.0 Phenotypic complexity (n) R el at ive co nve rg en ce ti me