The Arrival

of the

frequent

Outline

Evolution

 

The arrival of the frequent

 

Bias in genotype-phenotype maps

 

Origins of the bias

shaped by

needs

from

causes

genotype-phenotype maps

Wright-Fisher model

Evolution

Darwin's idea

(natural selection or survival of the fittest):

things that replicate themselves reliably stick around

Mendel's idea (genes):

in life, genes are the discrete units that replicate (i.e. hold heritable information)

Modern evolutionary synthesis:

Brings the two together into the powerful framework of population genetics.

Wright-Fisher model

The arrival of the frequent: background

Haploid, with mutations and selection

t=1
t=1t=1
t=2
t=2t=2
g_1
g1g_1
g_1
g1g_1
g_1
g1g_1
g_1
g1g_1
g_1
g1g_1
g_1
g1g_1
g_2
g2g_2
g_2
g2g_2
n_1
n1n_1
n_2
n2n_2
n_1+n_2 = N
n1+n2=Nn_1+n_2 = N

, number of individuals fixed

Individuals at new generation choose parent randomly from previous generation and inherit from it

g_1
g1g_1
g_1
g1g_1
g_1
g1g_1
g_1
g1g_1
g_1
g1g_1
g_2
g2g_2
g_2
g2g_2
g_2
g2g_2

we add mutations:

when copying the genotype of the parent:

for each one of the L letters,replace it by a random different letter with probability    

genotypes are strings of fixed length L over an alphabet with K letters

\mu
μ\mu

and selection:

selection coefficient     determines the fitness

fitness multiplies the probabilistic weight of an offspring choosing that individual as parent

P = \frac{1}{N}
P=1NP = \frac{1}{N}
P = \frac{1+s_1}{n_1 (1+s_1) + n_2 (1+s_2)}
P=1+s1n1(1+s1)+n2(1+s2)P = \frac{1+s_1}{n_1 (1+s_1) + n_2 (1+s_2)}
s
ss
1+s
1+s1+s

normalization

The fitness determines the relative number of offspring of different parents

\frac{\langle\text{number of offspring of parent 1}\rangle}{\langle\text{number of offspring of parent 2}\rangle} = \frac{1+s_1}{1+s_2}
⟨number of offspring of parent 1⟩⟨number of offspring of parent 2⟩=1+s11+s2\frac{\langle\text{number of offspring of parent 1}\rangle}{\langle\text{number of offspring of parent 2}\rangle} = \frac{1+s_1}{1+s_2}

Modern evolutionary synthesis argued for natural selection as the primary force shaping the outcomes of evolution

More recent developments and ideas suggest that developmental bias may also guide evolution.

One way to express developmental bias is via a genotype-phenotype map

One way this bias can affect evolution is via the arrival of the frequent

Genotype-phenotype maps

Mutations act on genotypes

Selection acts on phenotypes

The arrival of the frequent: background

From: Schuster, P. A testable genotype phenotype map: Modeling evolution of RNA molecules. In: Lassig, M. and Valleriani, A., editors, Biological Evolution and Statistical Physics, pp. 56{83. Springer-Verlag, Berlin, 2002.

Neutral evolution

Neutral spaces: connected networks of genotypes mapping to same phenotype

Genotype space, links represent single-point mutations

Neutral exploration: Evolution explores neutral space, being exposed to larger number of neighbouring possibilities, before switching to a different, better, phenotype

The arrival of the frequent: background

Arrival times

The arrival of the frequent

How many generations does it take to find a given phenotype p, starting from phenotype q?

m_p
mpm_p
q = \frac{m_p}{N}
q=mpNq = \frac{m_p}{N}

For each set of parents at one generation,

number of p-type offspring at the next generation

follows a Binomial distribution, because each offspring is i.i.d

with mean

, and probability of success

Mean field approximation:

replace        by an appropriate average

m_p
mpm_p

Probability at least one offpsring of type p:

\alpha = 1-(1-q)^{NT} = 1-(1-\frac{m_p}{N})^{NT} \approx 1-e^{-m_p T}
α=1−(1−q)NT=1−(1−mpN)NT≈1−e−mpT\alpha = 1-(1-q)^{NT} = 1-(1-\frac{m_p}{N})^{NT} \approx 1-e^{-m_p T}

Arrival times

The arrival of the frequent

Median arrival time:

T_{1/2} = \frac{\ln{2}}{m_p}
T1/2=ln2mpT_{1/2} = \frac{\ln{2}}{m_p}

          is calculated differently in different regimes. A particularly easy regime is the polymorphic limit, where the population is spread over the whole neutral space at each generation. Then

m_p
mpm_p
m_p = NL\mu \Phi_{pq}
mp=NLμΦpqm_p = NL\mu \Phi_{pq}

where,           is the probability that a one-point mutation leads to a phenotype p genotype, averaged over all genotypes in q's neutral space

\Phi_{pq}
Φpq\Phi_{pq}

The arrival of the frequent

Define the phenotype frequency         

as the fraction of genotypes that map to phenotype p.

F_p
FpF_p

If neutral spaces sufficiently large,                 

\Phi_{pq} \approx F_p
Φpq≈Fp\Phi_{pq} \approx F_p

Therefore, approximately, 

T_p \propto \frac{1}{F_p}
Tp∝1FpT_p \propto \frac{1}{F_p}

Frequent phenotypes arrive faster!

and therefore, often before unfrequent phenotypes

See more at: The Arrival of the Frequent: How Bias in Genotype-Phenotype Maps Can Steer Populations to Local Optima

The arrival of the frequent refers to the non-equilibrium phenomenon of arriving, but it can have very significant effects in evolutionary outcomes.

The arrival of the frequent

Phenotype 1 is more likely to fix 

because it will be "discovered" many more times

Eventually, p2 would arrive and fix, but the environment is likely to have changed by then.

Bias in GP maps

many GP maps have been found to show large biases towards a small number of phenotypes

Nature seems to have fixed on these few most frequent phenotypes

Other alternatives, maybe better ones, may just have never arrived.

 

The structure of the genotype–phenotype map strongly constrains the evolution of non-coding RNA

A tractable genotype–phenotype map modelling the self-assembly of protein quaternary structure

Images from:

Bias in GP maps

In the GP maps studied frequent (large neutral set) phenotypes also tend to have:

  • higher mutationally robustness
  • higher phenotypic evolvability

Surprisingly they also tend to be more simple, as measured by Kolmogorov complexity. This can also make the phenotype more environmentally robust.

All of these things are beneficial for an organism, so GP map bias may explain 

why evolution is so effective

Simplicity bias in GP maps

Example: Boolean network model of genetic regulatory networks

Complexity:

Image by Chico Calmargo

Origin of the bias

Idea 1: Genotype code has constrained and unconstrained parts

Simplest model:

Fibonacci GP map

Phenotypes with shorter contrained/coding part have larger neutral spaces

and are also more robust, evolvable, and are trivially simpler!

Origin of the bias

Idea 2: Shannon-Fano-Elias code and the coding theorem

Origin of the bias

Idea 3: Random finite state transducer

A way of describing computable maps,

with a simple parameter describing the complexity of the map: number of states

Origin of the bias

Random finite state transducers show bias!

At least simple ones (4 or 5 states in this case)

Origin of the bias

Bias in finite transducers: why?

\approx \frac{2000}{25}(\frac{1}{4})((2^9)+(2^8)+(2^7)+(2^6)+(2^5))
≈200025(14)((29)+(28)+(27)+(26)+(25))\approx \frac{2000}{25}(\frac{1}{4})((2^9)+(2^8)+(2^7)+(2^6)+(2^5))

How often do we expect                       or                       ?

111111111
000000000

Non-coding states -> cycles through non-coding states

\approx 2000((\frac{1}{5}+\frac{4}{25})(\frac{1}{4})(\frac{1}{2})(2^5+2^4+2^3+2^2+2^1))
≈2000((15+425)(14)(12)(25+24+23+22+21))\approx 2000((\frac{1}{5}+\frac{4}{25})(\frac{1}{4})(\frac{1}{2})(2^5+2^4+2^3+2^2+2^1))

How often do we expect                       or                       ?

101010101
010101010
Made with Slides.com